2512.23944

# Positive specializations of KK-theoretic Schur PP- and QQ-functions **Authors**: - Eric Marberg (Department of Mathematics) ## Abstract Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a $K$ -theoretic lift of the classical Edrei–Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse’s $K$ -theoretic Schur $P$ - and $Q$ -functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei–Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice. ## 1 Introduction This note extends Yeliussizov’s classification of the positive specializations of symmetric Grothendieck functions from [24] to their shifted analogues. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered variant of the shifted Young lattice. The rest of this introduction provides an outline of our main results. ### 1.1 Positive specializations Let $A$ be an (associative and unital) algebra defined over the field of real numbers $\mathbb{R}$ . A specialization of $A$ is an algebra morphism $\varphi:A\to\mathbb{R}$ . A specialization of $A$ is positive relative to a subset $S\subseteq A$ if $\varphi(s)\geq 0$ for all $s\in S$ . In this case we also say that $\varphi$ is $S$ -positive. Write $\mathbb{R}\llbracket\mathbf{x}\rrbracket=\mathbb{R}\llbracket x_{1},x_{2},x_{3},\dots\rrbracket$ for the ring of formal power series with real coefficients in a countable sequence of commuting variables. Recall that a partition is a weakly decreasing sequence of integers $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq 0)$ with finite sum $|\lambda|$ , and that $\mu\subseteq\lambda$ means that $\mu_{i}\leq\lambda_{i}$ for all $i$ . Fix a real number $\beta\in\mathbb{R}$ . The symmetric Grothendieck functions $G^{(\beta)}_{\lambda}$ (indexed by arbitrary partitions $\lambda$ [6, 11]) and their skew analogues $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ (indexed by pairs of partitions $\mu\subseteq\lambda$ [23]) are certain elements of $\mathbb{R}\llbracket\mathbf{x}\rrbracket$ that are symmetric under all permutations of the $x$ -variables. For the precise definition, see Section 3.1. When $\beta=-1$ , these power series are significant in $K$ -theory as representatives of the classes of the structure sheaves of Schubert varieties in the complex Grassmannian [2, §8]. When $\beta=0$ , we recover the usual (skew) Schur functions $s_{\lambda}=G^{(0)}_{\lambda}$ and $s_{\lambda/\mu}=G^{(0)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}.$ Let $\Gamma^{(\beta)}$ be the $\mathbb{R}$ -linear span of all skew symmetric Grothendieck functions $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ . The power series $G^{(\beta)}_{\lambda}=G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\emptyset}$ provide an $\mathbb{R}$ -basis for this vector space, which is actually a subalgebra of $\mathbb{R}\llbracket\mathbf{x}\rrbracket$ with unit $G^{(\beta)}_{0}=1$ and generators $G^{(\beta)}_{n}=G^{(\beta)}_{(n)}$ for $n\in\mathbb{Z}_{>0}$ [2]. We say that a specialization of $\Gamma^{(\beta)}$ is $G^{(\beta)}$ -positive if it is positive relative to the set of all functions $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ . Yeliussizov [23] has classified this set of specializations in the following way. Let $$ G_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=G^{(1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\quad\text{and}\quad\overline{G}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=G^{(-1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu} \tag{1} $$ and define $G_{\lambda}$ and $\overline{G}_{\lambda}$ analogously. Then write $\Gamma=\Gamma^{(1)}$ and $\overline{\Gamma}=\Gamma^{(-1)}$ for the $\mathbb{R}$ -bialgebras generated by these power series. As will be clarified in Section 3.2, to classify the $G^{(\beta)}$ -positive specializations of $\Gamma^{(\beta)}$ it suffices to identify the $G$ -positive specializations of $\Gamma$ , the $\overline{G}$ -positive specializations of $\overline{\Gamma}$ , and the Schur positive specializations of the algebra of bounded degree symmetric functions $\mathsf{Sym}=\Gamma^{(0)}$ . The solution to the third classification problem is well-known (see, e.g., [24, §1.2]) and will be reviewed in Section 2.2. In the following theorem, let $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ and $b=(b_{1}\geq b_{2}\geq\dots\geq 0)$ be sequences of nonnegative real numbers and let $\gamma\in\mathbb{R}_{\geq 0}$ . Then define $$ \textstyle C=C(a,b,\gamma)=e^{\gamma}\prod_{n=1}^{\infty}\frac{1+a_{n}}{1-b_{n}}. $$ When this infinite product converges, we necessarily have $\sum_{n=1}^{\infty}(a_{n}+b_{n})<\infty$ . **Theorem 1.1 ([24])** *An algebra morphism $\rho:\Gamma\to\mathbb{R}$ is a $G$ -positive specialization of $\Gamma$ if and only if for some choice of $a$ , $b$ , and $\gamma$ satisfying $\max(b)<1\leq C=1+\rho(G_{1})<\infty$ one has $$ \sum_{n\geq 0}\rho(G_{n}+G_{n+1})z^{n}=Ce^{\gamma z}\prod_{n=1}^{\infty}\frac{1+b_{n}z}{1-a_{n}z}. $$* This result is equivalent to an explicit formula for any $G$ -positive specialization of $\Gamma$ in terms of the parameters $a$ , $b$ , $\gamma$ , and $C$ ; see Section 3.3. Results in [24] give a similar (but slightly more subtle) classification of the $\overline{G}$ -positive specializations of $\overline{\Gamma}$ , which we discuss in Section 3.5. ### 1.2 Shifted variants The purpose of this note is to extend Theorem 1.1 and its signed analogue to shifted versions of the bialgebra $\Gamma^{(\beta)}$ . Recall that a partition $\lambda=(\lambda_{1}>\lambda_{2}>\dots\geq 0)$ is strict if its nonzero parts are all distinct. Ikeda and Naruse [9] introduced the $K$ -theoretic Schur $P$ - and $Q$ -functions $$ GP^{(\beta)}_{\lambda}\in\Gamma^{(\beta)}\quad\text{and}\quad GQ^{(\beta)}_{\lambda}\in\Gamma^{(\beta)} $$ for all strict partitions $\lambda$ . These have skew analogues $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\in\Gamma^{(\beta)}$ and $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\in\Gamma^{(\beta)}$ indexed by pairs of strict partitions $\mu\subseteq\lambda$ [13]. The precise definitions are reviewed in Section 3.1. When $\beta=-1$ , these power series are significant in $K$ -theory as representatives of the classes of the structure sheaves of Schubert varieties in the orthogonal and Lagrangian Grassmannians [9]. When $\beta=0$ , we recover the classical Schur $P$ and $Q$ -functions and their skew versions, namely: $$ P_{\lambda}=GP^{(0)}_{\lambda}\text{ and }Q_{\lambda}=GQ^{(0)}_{\lambda}\quad\text{along with}\quad P_{\lambda/\mu}=GP^{(0)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\text{ and }Q_{\lambda/\mu}=GQ^{(0)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}. \tag{0} $$ Let $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ be the respective $\mathbb{R}$ -vector spaces spanned by all $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ ’s and $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ ’s. These vector spaces turn out to be sub-algebras with bases $$ \Gamma_{Q}^{(\beta)}=\mathbb{R}\textnormal{-span}\left\{GQ^{(\beta)}_{\lambda}:\lambda\text{ strict}\right\}\subsetneq\Gamma_{P}^{(\beta)}=\mathbb{R}\textnormal{-span}\left\{GP^{(\beta)}_{\lambda}:\lambda\text{ strict}\right\}\subsetneq\Gamma^{(\beta)}\subsetneq\mathbb{R}\llbracket\mathbf{x}\rrbracket. $$ We say that specializations of $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ are $GP^{(\beta)}$ -positive and $GQ^{(\beta)}$ -positive if they are positive relative to the respective sets of skew functions $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ and $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ . To understand these specializations, it suffices to treat the cases when $\beta\in\{-1,0,1\}$ . The situation when $\beta=0$ is known [20] and will be reviewed in Section 2.3. For the other cases, let $$ GP_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=GP^{(1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu},\qquad GQ_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=GQ^{(1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu},\qquad\overline{GP}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=GP^{(-1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu},\qquad\overline{GQ}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=GQ^{(-1)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}, \tag{1} $$ and define $GP_{\lambda}$ , $\overline{GP}_{\lambda}$ , $GQ_{\lambda}$ , and $\overline{GQ}_{\lambda}$ likewise. Also let $$ \Gamma_{P}=\Gamma_{P}^{(1)}\quad\text{and}\quad\overline{\Gamma}_{P}=\Gamma_{P}^{(-1)}\quad\text{along with}\quad\Gamma_{Q}=\Gamma_{Q}^{(1)}\quad\text{and}\quad\overline{\Gamma}_{Q}=\Gamma_{Q}^{(-1)}. \tag{1} $$ We define specializations of these algebras to be $GP$ -, $\overline{GP}$ -, $GQ$ -, or $\overline{GQ}$ -positive in the obvious way. We may now present our first new theorem. Let $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ be a sequence of nonnegative real numbers and suppose $\gamma\in\mathbb{R}_{\geq 0}$ . Define $\overline{x}=\frac{-x}{1+x}$ for any parameter $x$ and let $$ \textstyle D=D(a,\gamma)=e^{\gamma}\prod_{n=1}^{\infty}\frac{1-\overline{a_{n}}}{1-a_{n}}. $$ When this infinite product converges, we necessarily have $\sum_{n=1}^{\infty}a_{n}<\infty$ . **Theorem 1.2** *The $GP$ -positive specializations of $\Gamma_{P}$ and the $GQ$ -positive specializations of $\Gamma_{Q}$ are each obtained by restricting some $G$ -positive specialization of $\Gamma$ . More specifically, if $\rho:\Gamma\to\mathbb{R}$ is an algebra morphism then the following properties are equivalent: 1. $\rho$ restricts to a $GP$ -positive specialization of $\Gamma_{P}$ . 1. $\rho$ restricts to a $GQ$ -positive specialization of $\Gamma_{Q}$ . 1. For some choice of $a$ and $\gamma$ satisfying $1\leq D=1+\rho(GP_{1})<\infty$ one has $$ \sum_{n\geq 0}\rho(GQ_{n}+GQ_{n+1})z^{n}=D^{2}e^{2\gamma z}\prod_{n=1}^{\infty}\frac{1-\overline{a_{n}}z}{1-a_{n}z}. $$* The equivalence of (a) and (b) is surprising, since negative coefficients are required to express an arbitrary $GQ$ -function as a linear combination of $GP$ -functions [4]. As with Theorem 1.1, this result is equivalent to an explicit formula for any $GP$ - or $GQ$ -positive specialization of $\Gamma_{P}$ or $\Gamma_{Q}$ ; see Section 3.3. We also derive a similar classification of the $\overline{GP}$ and $\overline{GQ}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ in Section 3.5. ### 1.3 Applications to harmonic functions The $\beta=1$ classification results presented in this introduction are more natural algebraically than their $\beta=-1$ versions, since if $\beta\geq 0$ then we have $$ G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\in\mathbb{R}\textnormal{-span}\left\{G^{(\beta)}_{\nu}\right\},\quad GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\in\mathbb{R}\textnormal{-span}\left\{GP^{(\beta)}_{\nu}\right\},\quad GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\in\mathbb{R}\textnormal{-span}\left\{GQ^{(\beta)}_{\nu}\right\} $$ by results in [2, 17]; see the discussion in Section 3.2. Hence, a specialization is $G$ -, $GP$ -, or $GQ$ -positive if and only if it is positive relative to the defining basis of $\Gamma$ , $\Gamma_{P}$ , or $\Gamma_{Q}$ . Additionally, Yeliussizov has shown [24, Thm. 5.2] that the subset of $G$ -positive specializations $\rho$ of $\Gamma$ that are normalized in the sense that $\rho(G_{(1)})=1$ are naturally in bijection with the extreme points of the convex set of harmonic functions on a certain filtered Young graph. In Section 3.6, we prove an extension of this result that relates normalized $GP$ -positive specializations of $\Gamma_{P}^{(\beta)}$ to the extreme points of the convex set of harmonic functions on a shifted filtered Young graph. ### Acknowledgments The author thanks Alimzhan Amanov, Joel Lewis, Damir Yeliussizov for helpful discussions. This article is based on work supported by the National Science Foundation under grant DMS-1929284 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Categorification and Computation in Algebraic Combinatorics semester program. The author was also partially supported by Hong Kong RGC grants 16304122 and 16304625. ## 2 Preliminaries This section contains some general background material on specializations of bialgebras and classical algebras of symmetric functions. ### 2.1 Bialgebra specializations We assume some familiarity with the definition of a bialgebra and its basic properties. A good reference for this material is [7]. Suppose $B$ is a bialgebra defined over $\mathbb{R}$ with coproduct $\Delta_{B}$ . Let $\otimes$ be the tensor product over $\mathbb{R}$ and write $\nabla_{\mathbb{R}}:\mathbb{R}\otimes\mathbb{R}\to\mathbb{R}$ for the usual multiplication map. **Definition 2.1** *The union of two specializations $\rho_{1},\rho_{2}:B\to\mathbb{R}$ is the specialization $$ \rho_{1}\sqcup\rho_{2}=\nabla_{\mathbb{R}}\circ(\rho_{1}\otimes\rho_{2})\circ\Delta_{B}. $$* As $\Delta_{B}$ is associative, it holds that $\rho_{1}\sqcup(\rho_{2}\sqcup\rho_{3})=(\rho_{1}\sqcup\rho_{2})\sqcup\rho_{3}$ so we many consider iterated unions and omit all parentheses in expressions for these. In general the union operation on specializations may not be commutative, but if $B$ is cocommutative then $\rho_{1}\sqcup\rho_{2}=\rho_{2}\sqcup\rho_{1}$ . **Definition 2.2** *A subset $S\subseteq B$ is comultiplicative if $$ \Delta_{B}(s)\in\mathbb{R}_{\geq 0}\textnormal{-span}\{s_{1}\otimes s_{2}:s_{1},s_{2}\in S\}\quad\text{for all $s\in S$.} $$* If $S$ has this the property then any union of $S$ -positive specializations is also $S$ -positive. Hence, if $S$ is comultiplicative then the set of $S$ -positive specializations of $B$ form a semigroup. ### 2.2 Symmetric functions Let $\mathsf{Sym}=\mathsf{Sym}(\mathbf{x})\subset\mathbb{R}\llbracket\mathbf{x}\rrbracket$ denote the ring bounded degree symmetric functions [15, §I]. We may express $\mathsf{Sym}$ as a polynomial ring in two ways as $\mathsf{Sym}=\mathbb{R}[h_{1},h_{2},h_{3},\dots]=\mathbb{R}[e_{1},e_{2},e_{3},\dots]$ where $h_{n}$ and $e_{n}$ are the complete homogeneous and elementary symmetric functions $$ \textstyle h_{n}=\sum_{1\leq i_{1}\leq i_{2}\leq\dots\leq i_{n}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}\quad\text{and}\quad e_{n}=\sum_{1\leq i_{1}<i_{2}<\dots<i_{n}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}. $$ The $\mathbb{R}$ -algebra $\mathsf{Sym}$ has a bialgebra structure [7, §2] in which the counit $\varepsilon:\mathsf{Sym}\to\mathbb{R}$ is the map setting $x_{1}=x_{2}=x_{3}=\dots=0$ and where the coproduct $\Delta:\mathsf{Sym}\otimes\mathsf{Sym}\to\mathsf{Sym}$ satisfies $$ \textstyle\Delta(h_{n})=\sum_{i=0}^{n}h_{i}\otimes h_{n-i}\quad\text{and}\quad\Delta(e_{n})=\sum_{i=0}^{n}e_{i}\otimes e_{n-i}\quad\text{for all $n\in\mathbb{Z}_{\geq 0}$}. $$ In this formula, we set $e_{0}=h_{0}=1$ . The coproduct for $\mathsf{Sym}$ may be computed by replacing the variables $x_{1},x_{2},\dots$ with a doubled sequence $x_{1},x_{2},\dots,y_{1},y_{2},\dots$ and then applying the natural isomorphism $\mathsf{Sym}(\mathbf{x},\mathbf{y})\xrightarrow{\sim}\mathsf{Sym}\otimes\mathsf{Sym}$ . There is a unique bialgebra involution $\omega:\mathsf{Sym}\to\mathsf{Sym}$ with $$ \omega(h_{n})=e_{n}\quad\text{and}\quad\omega(e_{n})=h_{n}\quad\text{for all $n\in\mathbb{Z}_{\geq 0}$.} $$ The cocommutative bialgebra $\mathsf{Sym}$ is graded and connected, and therefore is a Hopf algebra. Its antipode is the composition of $\omega$ with the evaluation map $x_{i}\mapsto-x_{i}$ negating all variables [7, §2]. The Hopf algebra $\mathsf{Sym}$ has a distinguished $\mathbb{R}$ -basis of Schur functions $s_{\lambda}$ indexed by all partitions, which have a skew generalization $s_{\lambda/\mu}$ indexed by pairs of partitions $\mu\subseteq\lambda$ . A succinct combinatorial definition is provided by the formulas $s_{\lambda}=s_{\lambda/\emptyset}$ and $s_{\lambda/\mu}=\sum_{T}x^{T}$ , where the sum is over all semistandard tableaux of shape $\lambda/\mu$ [15, §I.5]. We refer to the specializations of $\mathsf{Sym}$ that are positive with respect to the set of all skew Schur functions $s_{\lambda/\mu}$ as Schur positive. This is the same as the set of specializations that are positive with respect to the basis of Schur functions, since each skew Schur function is a $\mathbb{Z}_{\geq 0}$ -linear combination of ordinary Schur functions [15, §I.5, (5.3)]. The set of skew Schur functions is comultiplicative since $\Delta(s_{\lambda/\mu})=\sum_{\mu\subseteq\kappa\subseteq\lambda}s_{\kappa/\mu}\otimes s_{\lambda/\kappa}$ [15, §I.5, (5.10)] and so the set of Schur positive specializations is a commutative semigroup under the union operation. These specializations have a well-known classification, which we briefly review. **Definition 2.3** *For an infinite sequence of nonnegative real numbers $a=(a_{1}\geq a_{2}\geq a_{3}\geq\dots\geq 0)$ with finite sum, let $\phi_{a}:\mathsf{Sym}\to\mathbb{R}$ and $\varepsilon_{a}:\mathsf{Sym}\to\mathbb{R}$ be the maps with the formulas $$ \phi_{a}(f)=f(a_{1},a_{2},a_{3},\dots)\quad\text{and}\quad\varepsilon_{a}=\phi_{a}\circ\omega. $$* When $a$ is a single real number, let $\phi_{a}=\phi_{(a,0,0,0,\dots)}$ and when $a=(a_{1}\geq a_{2}\geq\dots\geq a_{k})$ is a finite sequence of real numbers, let $\phi_{a}=\phi_{(a_{1},a_{2},\dots,a_{k},0,0,0,\dots)}$ . Extend the definition of $\varepsilon_{a}$ similarly. **Definition 2.4** *For any real number $\gamma\geq 0$ write $\pi_{\gamma}:\mathsf{Sym}\to\mathbb{R}$ for the unique algebra morphism satisfying $\pi_{\gamma}(h_{n})=\tfrac{\gamma^{n}}{n!}$ for all $n\in\mathbb{Z}_{\geq 0}$ .* By [24, Lem. 6.5] it holds for any $f\in\mathsf{Sym}$ that $$ \displaystyle\pi_{\gamma}(f)=\lim_{N\to\infty}f(\underbrace{\gamma/N,\gamma/N,\dots,\gamma/N}_{N\text{ terms}},0,0,0,\dots). $$ Using this, one checks that we also have $\pi_{\gamma}(e_{n})=\tfrac{\gamma^{n}}{n!}$ . Hence $$ \pi_{\gamma}\circ\omega=\pi_{\gamma}. $$ The following result is equivalent to the classical Edrei–Thoma theorem [5, 22]. **Theorem 2.5 (Edei–Thoma; see[24, Thm. 2.4])** *The Schur positive specializations of $\mathsf{Sym}$ are the maps $\rho=\phi_{\alpha}\sqcup\varepsilon_{b}\sqcup\pi_{c}$ where $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ and $b=(b_{1}\geq b_{2}\geq\dots\geq 0)$ have finite sum and $\gamma\in\mathbb{R}_{\geq 0}$ . The representation of $\rho$ is unique and it holds that $\textstyle\rho(h_{1})=\sum_{n=1}^{\infty}(a_{n}+b_{n})+\gamma_{n}.$* Let $z$ be a formal variable. Then write $H(z),E(z)\in\mathsf{Sym}\llbracket z\rrbracket$ for the power series $$ H(z):=\sum_{n\geq 0}h_{n}z^{n}=\prod_{n\geq 1}\frac{1}{1-x_{n}z}\quad\text{and}\quad E(z):=\sum_{n\geq 0}e_{n}z^{n}=\prod_{n\geq 1}(1+x_{n}z)=H(-z)^{-1}. $$ Any specialization $\rho:\mathsf{Sym}\to\mathbb{R}$ extends to an algebra morphism $\mathsf{Sym}\llbracket z\rrbracket\to\mathbb{R}\llbracket z\rrbracket$ with the formula $\textstyle\rho\left(\sum_{n\geq 0}f_{n}z^{n}\right)=\sum_{n\geq 0}\rho(f_{n})z^{n}.$ Under this convention, since $\omega(H(z))=E(z)$ , one has $$ \textstyle\phi_{a}(H(z))=\prod_{n\geq 1}\frac{1}{1-a_{n}z},\quad\varepsilon_{a}(H(z))=\prod_{n\geq 1}(1+a_{n}z),\quad\text{and}\quad\pi_{\gamma}(H(z))=e^{\gamma z} $$ as well as $$ \textstyle\phi_{a}(E(z))=\prod_{n\geq 1}(1+a_{n}z),\quad\varepsilon_{a}(E(z))=\prod_{n\geq 1}\frac{1}{1-a_{n}z},\quad\text{and}\quad\pi_{\gamma}(E(z))=e^{\gamma z}. $$ The coproduct formula (2.3) implies for any algebra morphisms $\rho_{i}:\mathsf{Sym}\to\mathbb{R}$ that $$ (\rho_{1}\sqcup\rho_{2})(H(z))=\rho_{1}(H(z))\rho_{2}(H(z))\quad\text{and}\quad(\rho_{1}\sqcup\rho_{2})(E(z))=\rho_{1}(E(z))\rho_{2}(E(z)) $$ so we have $\rho=\phi_{\alpha}\sqcup\varepsilon_{b}\sqcup\pi_{\gamma}$ if and only if $\rho(H(z))=e^{\gamma z}\prod_{n\geq 1}\frac{1+b_{n}z}{1-a_{n}z}.$ Moreover, we see that $$ \phi_{a}=\phi_{a_{1}}\sqcup\phi_{a_{2}}\sqcup\phi_{a_{3}}\sqcup\cdots\quad\text{and}\quad\varepsilon_{a}=\varepsilon_{a_{1}}\sqcup\varepsilon_{a_{2}}\sqcup\varepsilon_{a_{3}}\sqcup\cdots. $$ ### 2.3 Shifted symmetric functions Let $\mathsf{SSym}$ be the subspace of power series $f\in\mathsf{Sym}$ satisfying the supersymmetry property $$ f(-z,z,x_{1},x_{2},x_{3},\dots)\in\mathbb{R}\llbracket x_{1},x_{2},x_{3},\dots\rrbracket. $$ This subspace is a Hopf subalgebra of $\mathsf{Sym}$ , and has a basis given by the Schur $P$ -functions $P_{\lambda}$ indexed by all strict partitions $\lambda$ ; see [15, §III.8]. Since the scalar field is $\mathbb{R}$ , the Schur $Q$ -functions $Q_{\lambda}=2^{\ell(\lambda)}P_{\lambda}$ form a second basis. The power series $P_{\lambda}$ and $Q_{\lambda}$ are obtained from the symmetric functions $GP^{(\beta)}_{\lambda}$ and $GQ^{(\beta)}_{\lambda}$ defined in Section 3.1 by setting $\beta=0$ . Any specialization of $\mathsf{SSym}$ that is positive with respect to the basis $\{P_{\lambda}\}$ is clearly also positive with respect to the basis $\{Q_{\lambda}\}$ . The classification of such Schur $P$ -positive specializations is also known [20]. For $n\in\mathbb{Z}_{\geq 0}$ define $q_{n}=\sum_{i+j=n}e_{i}h_{j}$ so that $$ \textstyle q_{0}=1,\quad q_{1}=2h_{1},\quad\text{and}\quad Q(z):=\sum_{n\in\mathbb{Z}_{\geq 0}}q_{n}z^{n}=E(z)H(z)=\prod_{n\geq 1}\frac{1+x_{n}z}{1-x_{n}z}. $$ The elements $\{q_{n}\}$ generate $\mathsf{SSym}$ as an $\mathbb{R}$ -algebra and are algebraically independent [15, §III.8]. **Theorem 2.6 (Nazarov[20])** *A specialization $\rho:\mathsf{SSym}\to\mathbb{R}$ is Schur $P$ -positive if and only if there are real numbers $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ and $\gamma\geq 0$ with $\sum_{n=1}^{\infty}\alpha_{n}<\infty$ such that $$ \sum_{n\geq 0}\rho(q_{n})z^{n}=e^{2\gamma z}\prod_{n\geq 1}\frac{1+a_{n}z}{1-a_{n}z}. $$ In this case $\textstyle\rho(h_{1})=\tfrac{1}{2}\rho(q_{1})=\sum_{n=1}^{\infty}a_{n}+\gamma$ and $\rho$ coincides with $\phi_{a}\sqcup\pi_{\gamma}$ restricted to $\mathsf{SSym}.$* One can derive this theorem in a direct algebraic way from Theorem 2.5. The argument is essentially the same as the proof of Theorem 3.17, just changing various constructions and properties involving $\beta=1$ to their simplified versions with $\beta=0$ . ## 3 Positive K-theoretic specializations This section contains our main results and applications. The first two subsections review the precise definitions of $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ , $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ , and $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ , and the algebraic structures they generate. Our main new results are Theorems 3.17 and 3.29 and Corollary 3.38. ### 3.1 Set-valued tableaux Let $D\subseteq E$ be finite subsets of $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ and define the interior of $D$ to be the set $$ \mathrm{Int}(D)=\{(i,j)\in D:(i+1,j)\in D\text{ or }(i,j+1)\in D\}. $$ For any finite sets $A,B\subset\mathbb{R}$ , write $A\preceq B$ if either set is empty or if $\max(A)\leq\min(B)$ . **Definition 3.1** *A set-valued tableau of shape $E/\hskip-2.84526pt/D$ is a map $T$ that assigns a finite set $T_{ij}\subset\mathbb{R}_{>0}$ to each $(i,j)\in E\setminus\mathrm{Int}(D)$ such that $T_{ij}$ is nonempty if $(i,j)\in E\setminus D$ and such that $$ T_{ij}\preceq T_{i,j+1}\quad\text{and}\quad T_{ij}\preceq T_{i+1,j}\quad\text{for all $(i,j)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$} $$ under that convention that $T_{ij}=\varnothing$ if $(i,j)\notin E\setminus\mathrm{Int}(D)$ . For any such tableau $T$ define $$ \textstyle|T|=\sum_{(i,j)\in E\setminus\mathrm{Int}(D)}|T_{ij}|\quad\text{and}\quad x^{T}=\prod_{(i,j)\in E\setminus\mathrm{Int}(D)}\prod_{a\in T_{ij}}x_{\lceil a\rceil}. $$* **Example 3.2** *Suppose $E=D\sqcup\{(1,8),(1,9),(2,6),(3,5),(3,6),(4,4)\}$ where $$ D=\{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4)\}. $$ In the following picture, the positions indicated by “ $\cdot$ ” represent the elements of $\mathrm{Int}(D)$ , while the boxes and make up $D\setminus\mathrm{Int}(D)$ and $E\setminus D$ , respectively: $$ {\hbox{$\vbox{\hbox{\vtop{\halign{&\opttoksa@YT={\font@YT}\getcolor@YT{\save@YT{\opttoksb@YT}}\nil@YT\getcolor@YT{\startbox@@YT\the\opttoksa@YT\the\opttoksb@YT}#\endbox@YT\cr\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$\cdot$\hss}\vss}\kern-11.75085pt\vrule width=0.0pt,height=11.75085pt\kern 0.36993pt\kern 11.38092pt\vrule 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Choose partitions $\mu\subseteq\lambda$ and recall that the diagram of $\lambda$ is the set $$ \mathsf{D}_{\lambda}=\{(i,j):i\in\mathbb{Z}_{>0}\text{ and }1\leq\lambda_{j}\leq i\}. $$ Let $\mathsf{SVT}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)$ be the collection of set-valued tableaux $T$ of shape $\mathsf{D}_{\lambda}{/\penalty 50\hskip-2.84526pt/\penalty 50}\mathsf{D}_{\mu}$ whose values $T_{ij}$ are all sets of positive integers. **Definition 3.3 ([2])** *The symmetric Grothendieck function of shape $\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu$ is $$ \textstyle G^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}=\sum_{T\in\mathsf{SVT}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)}\beta^{|T|-|\lambda/\mu|}x^{T}\in\mathbb{R}\llbracket\mathbf{x}\rrbracket. $$ We also define $G^{(\beta)}_{\lambda}=G^{(\beta)}_{\lambda/\hskip-2.84526pt/\emptyset}$ and for convenience let $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=0$ when $\mu\not\subseteq\lambda$ .* **Example 3.4** *If $0<m\neq n$ then $G^{(\beta)}_{(n)/\hskip-2.84526pt/(m)}=G^{(\beta)}_{(n-m)}+\beta G^{(\beta)}_{(n-m+1)}$ where we set $G^{(\beta)}_{(0)}=1$ .* Now suppose $\mu\subseteq\lambda$ are strict partitions and recall that the shifted diagram of $\lambda$ is the set $$ \mathsf{SD}_{\lambda}=\{(i,i+j-1):(i,j)\in\mathsf{D}_{\lambda}\}. $$ Let $\mathsf{ShSVT}_{Q}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)$ be the set of set-valued tableaux $T$ of shape $\mathsf{SD}_{\lambda}{/\penalty 50\hskip-2.84526pt/\penalty 50}\mathsf{SD}_{\mu}$ with $$ T_{ij}\subset\tfrac{1}{2}\mathbb{Z}_{>0}\quad\text{and}\quad T_{ij}\cap T_{i,j+1}\subset\mathbb{Z}_{>0}\quad\text{and}\quad T_{ij}\cap T_{i+1,j}\subset\mathbb{Z}_{>0}-\tfrac{1}{2} $$ for all $(i,j)\in\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ . Also define $$ \mathsf{ShSVT}_{P}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)=\left\{T\in\mathsf{ShSVT}_{Q}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu):T_{ij}\subset\mathbb{Z}_{>0}\text{ whenever }i=j\right\}. $$ **Definition 3.5 ([9,13])** *The $K$ -theoretic Schur $Q$ -function of shape $\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu$ is $$ \textstyle GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}=\sum_{T\in\mathsf{ShSVT}_{Q}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)}\beta^{|T|-|\lambda/\mu|}x^{T}\in\mathbb{R}\llbracket\mathbf{x}\rrbracket. $$ Similarly, the $K$ -theoretic Schur $P$ -function of shape $\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu$ is $$ \textstyle GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}=\sum_{T\in\mathsf{ShSVT}_{P}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)}\beta^{|T|-|\lambda/\mu|}x^{T}\in\mathbb{R}\llbracket\mathbf{x}\rrbracket. $$ Let $GP^{(\beta)}_{\lambda}=GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\emptyset}$ and $GQ^{(\beta)}_{\lambda}=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\emptyset}$ and define $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}=0$ when $\mu\not\subseteq\lambda$ .* We abbreviate by setting $$ G^{(\beta)}_{0}=GP^{(\beta)}_{0}=GQ^{(\beta)}_{0}=1,\quad G^{(\beta)}_{n}=G^{(\beta)}_{(n)},\quad GP^{(\beta)}_{n}=GP^{(\beta)}_{(n)},\quad\text{and}\quad GQ^{(\beta)}_{n}=GQ^{(\beta)}_{(n)} $$ for $n\in\mathbb{Z}_{>0}$ . Notice that $\textstyle G^{(\beta)}_{1}=GP^{(\beta)}_{1}=\sum_{n\geq 1}e_{n}=E(1)-1.$ One also has [4] $$ GQ^{(\beta)}_{n}=2GP^{(\beta)}_{n}+\beta GP^{(\beta)}_{n+1}\quad\text{for all }n\in\mathbb{Z}_{>0}. $$ In general, $GQ^{(\beta)}_{\lambda}$ is an $\mathbb{R}$ -linear but not always an $\mathbb{R}_{\geq 0}$ -linear combination of $GP^{(\beta)}$ -functions [4]. **Example 3.6** *If $0<m\neq n$ then $GQ^{(\beta)}_{(n)/\hskip-2.84526pt/(m)}=GQ^{(\beta)}_{n-m}+\beta GQ^{(\beta)}_{n-m+1}$ while $$ GP^{(\beta)}_{(n)/\hskip-2.84526pt/(m)}=GQ^{(\beta)}_{n-m}+\begin{cases}\beta GQ^{(\beta)}_{n-m+1}&\text{if }m>1\\ \beta GP^{(\beta)}_{n}&\text{if }m=1.\end{cases} $$* As a variant of the above constructions, for any strict partitions $\mu\subseteq\lambda$ let $$ \mathsf{ShSVT}_{P}(\lambda/\mu)\subset\mathsf{ShSVT}_{P}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu)\quad\text{and}\quad\mathsf{ShSVT}_{Q}(\lambda/\mu)\subset\mathsf{ShSVT}_{Q}(\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu) $$ be the subsets of tableaux $T$ that have $T_{ij}=\varnothing$ for all $(i,j)\in\mathsf{SD}_{\mu}$ . **Definition 3.7 ([13])** *The $K$ -theoretic Schur $P$ - and $Q$ -functions of shape $\lambda/\mu$ are $$ GP^{(\beta)}_{\lambda/\mu}=\sum_{T\in\mathsf{ShSVT}_{P}(\lambda/\mu)}\beta^{|T|-|\lambda/\mu|}x^{T}\quad\text{and}\quad GQ^{(\beta)}_{\lambda/\mu}=\sum_{T\in\mathsf{ShSVT}_{Q}(\lambda/\mu)}\beta^{|T|-|\lambda/\mu|}x^{T}. $$* These power series are related to $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ and $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ by the identities $$ GP^{(\beta)}_{\lambda/\mu}=\sum_{\nu\subseteq\mu}(-\beta)^{|\mu|-|\nu|}GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\nu}\quad\text{and}\quad GQ^{(\beta)}_{\lambda/\mu}=\sum_{\nu\subseteq\mu}(-\beta)^{|\mu|-|\nu|}GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\nu} $$ where the sums are indexed by strict partitions [13, Cor. 5.7]. ### 3.2 Unbounded symmetric functions Let $\mathfrak{m}\mathsf{Sym}\subset\mathbb{R}\llbracket\mathbf{x}\rrbracket$ be the subalgebra of all symmetric formal power series, not necessarily of bounded degree. Each element of $\mathfrak{m}\mathsf{Sym}$ can be uniquely expressed as an infinite linear combination of Schur functions $s_{\lambda}$ , and we define $\mathfrak{m}\mathsf{Sym}\mathbin{\hat{\otimes}}\mathfrak{m}\mathsf{Sym}$ to be the real vector space of infinite $\mathbb{R}$ -linear combinations of tensor $s_{\lambda}\otimes s_{\mu}$ . This space is larger than the usual tensor product $\mathfrak{m}\mathsf{Sym}\otimes\mathfrak{m}\mathsf{Sym}$ . Each $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ is a finite $\mathbb{R}$ -linear combination of $G^{(\beta)}_{\nu}$ ’s [2] and each $G^{(\beta)}_{\lambda}$ is an infinite $\mathbb{R}$ -linear combination of Schur functions $s_{\mu}$ [12]. Hence, the $\mathbb{R}$ -vector space $\Gamma^{(\beta)}$ spanned by all $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ ’s has $$ \Gamma^{(\beta)}\subsetneq\mathfrak{m}\mathsf{Sym}\quad\text{and}\quad\Gamma^{(\beta)}\otimes\Gamma^{(\beta)}\subsetneq\mathfrak{m}\mathsf{Sym}\mathbin{\hat{\otimes}}\mathfrak{m}\mathsf{Sym}. $$ Similarly, when $\lambda$ and $\mu$ are strict partitions, each $GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ (respectively, $GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ ) is a finite $\mathbb{R}$ -linear combination of $GP^{(\beta)}_{\nu}$ ’s (respectively, $GQ^{(\beta)}_{\nu}$ ’s) [16, Prop. 4.33]. In turn, each $GQ^{(\beta)}_{\nu}$ is a finite linear combination of $GP^{(\beta)}$ -functions [4] and each $GP^{(\beta)}_{\nu}$ is a finite linear combination of $G^{(\beta)}$ -functions [18, Thm. 3.27]. Hence the vector spaces $\Gamma_{P}^{(\beta)}=\mathbb{R}\textnormal{-span}\left\{GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}:\text{strict }\mu\subseteq\lambda\right\}$ and $\Gamma_{Q}^{(\beta)}=\mathbb{R}\textnormal{-span}\left\{GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}:\text{strict }\mu\subseteq\lambda\right\}$ from the introduction satisfy $$ \Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\subsetneq\Gamma^{(\beta)}\quad\text{and}\quad\Gamma_{Q}^{(\beta)}\otimes\Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\otimes\Gamma_{P}^{(\beta)}\subsetneq\mathfrak{m}\mathsf{Sym}\mathbin{\hat{\otimes}}\mathfrak{m}\mathsf{Sym}. $$ Somewhat surprisingly, the subspaces $\Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\subsetneq\Gamma^{(\beta)}$ are subalgebras of $\mathfrak{m}\mathsf{Sym}$ [2, 14]. **Remark 3.8** *Let $\mathfrak{r}:\mathbb{R}\llbracket\mathbf{x}\rrbracket\to\mathbb{R}\llbracket\mathbf{x}\rrbracket$ be the rescaling morphism $f\mapsto f(|\beta|x_{1},|\beta|x_{2},|\beta|x_{3},\dots)$ . 1. First assume $\beta>0$ . Then we have $$ \mathfrak{r}(G_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu})=\beta^{|\lambda|-|\mu|}G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu},\ \ \mathfrak{r}(GP_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu})=\beta^{|\lambda|-|\mu|}GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu},\ \ \mathfrak{r}(GQ_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu})=\beta^{|\lambda|-|\mu|}GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}, $$ and so $\mathfrak{r}$ restricts to algebra isomorphisms $\Gamma\xrightarrow{\sim}\Gamma^{(\beta)}$ and $\Gamma_{P}\xrightarrow{\sim}\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}\xrightarrow{\sim}\Gamma_{Q}^{(\beta)}$ . Composition with $\mathfrak{r}$ gives a bijection from $G^{(\beta)}$ - to $G$ -positive specializations, as well as from $GP^{(\beta)}$ - to $GP$ -positive specializations and from $GQ^{(\beta)}$ - to $GQ$ -positive specializations. 1. Similarly, when $\beta<0$ the map $\mathfrak{r}$ restricts to algebra isomorphisms $\overline{\Gamma}\xrightarrow{\sim}\Gamma^{(\beta)}$ and $\overline{\Gamma}_{P}\xrightarrow{\sim}\Gamma_{P}^{(\beta)}$ and $\overline{\Gamma}_{Q}\xrightarrow{\sim}\Gamma_{Q}^{(\beta)}$ , and composition with $\mathfrak{r}$ gives a bijection from $G^{(\beta)}$ -, $GP^{(\beta)}$ -, and $GQ^{(\beta)}$ -positive specializations to $\overline{G}$ -, $\overline{GP}$ -, and $\overline{GQ}$ -positive specializations 1. Finally if $\beta=0$ then $\Gamma^{(\beta)}=\mathsf{Sym}$ and $\Gamma_{P}^{(\beta)}=\Gamma_{Q}^{(\beta)}=\mathsf{SSym}$ from Section 2.3. Thus, to classify all $G^{(\beta)}$ -, $GP^{(\beta)}$ -, and $GQ^{(\beta)}$ -positive specializations of our three $\mathbb{R}$ -algebras, one just needs to understand the classical case when $\beta=0$ and the two other cases when $\beta=\pm 1$ .* A linear map $f$ with domain $\mathfrak{m}\mathsf{Sym}$ is continuous if $f\left(\sum_{\lambda}c_{\lambda}s_{\lambda}\right)=\sum_{\lambda}c_{\lambda}f(s_{\lambda})$ for all $c_{\lambda}\in\mathbb{R}$ . The counit and coproduct of $\mathsf{Sym}$ extend to continuous linear maps $\mathfrak{m}\mathsf{Sym}\to\mathbb{R}$ and $\mathfrak{m}\mathsf{Sym}\to\mathfrak{m}\mathsf{Sym}\mathbin{\hat{\otimes}}\mathfrak{m}\mathsf{Sym}$ . Restricting these makes $\Gamma^{(\beta)}$ into a cocommutative bialgebra [2, Cor. 6.7] with $$ \Delta(G^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=\sum_{\mu\subseteq\kappa\subseteq\lambda}G^{(\beta)}_{\kappa/\hskip-2.84526pt/\mu}\otimes G^{(\beta)}_{\lambda/\hskip-2.84526pt/\kappa}. $$ Additionally, the subspaces $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ are sub-bialgebras of $\Gamma^{(\beta)}$ with coproduct formulas $$ \Delta(GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=\sum_{\mu\subseteq\kappa\subseteq\lambda}GP^{(\beta)}_{\kappa/\hskip-2.84526pt/\mu}\otimes GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\kappa}\quad\text{and}\quad\Delta(GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=\sum_{\mu\subseteq\kappa\subseteq\lambda}GQ^{(\beta)}_{\kappa/\hskip-2.84526pt/\mu}\otimes GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\kappa} $$ where the sums are indexed by strict partitions [16, Prop. 4.33]. **Remark 3.9** *In view of the above formulas, the generating sets $\{G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\}$ , $\{GP^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\}$ , and $\{GQ^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}\}$ are comultiplicative subsets of $\Gamma^{(\beta)}$ , $\Gamma_{P}^{(\beta)}$ , and $\Gamma_{Q}^{(\beta)}$ . Hence, the sets of $G^{(\beta)}$ -, $GP^{(\beta)}$ -, and $GQ^{(\beta)}$ -positive specializations for each bialgebra form commutative semigroups under the union operation.* We mention two other notable facts. First, each $GP^{(\beta)}_{\lambda}$ and $GQ^{(\beta)}_{\lambda}$ is a $\mathbb{Z}_{\geq 0}[\beta]$ -linear combination of $G^{(\beta)}_{\nu}$ ’s [18, Thms. 3.27 and 3.40]. Second, the multiplicative and comultiplicative structure constants for the three bialgebras $\Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\subsetneq\Gamma^{(\beta)}$ all belong to $\mathbb{Z}_{\geq 0}[\beta]$ ; see [2, Cors. 5.5 and 6.7] for $\Gamma^{(\beta)}$ and [14, Thm. 1.6] and [17, Cor. 4.42 and Rem. 4.43] for $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ . This implies: **Proposition 3.10** *Suppose $\beta\geq 0$ and $f$ , $g$ , and $h$ are specializations of $G^{(\beta)}$ , $GP^{(\beta)}$ , and $GQ^{(\beta)}$ . Then these specializations are respectively $G^{(\beta)}$ -, $GP^{(\beta)}$ -, and $GQ^{(\beta)}$ -positive if and only if $$ f(G^{(\beta)}_{\lambda})\geq 0,\quad g(GP^{(\beta)}_{\mu})\geq 0,\quad\text{and}\quad h(GQ^{(\beta)}_{\mu})\geq 0 $$ for all partitions $\lambda$ and all strict partitions $\mu$ . Moreover, if $f$ is $G^{(\beta)}$ -positive then its restrictions to $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ are respectively $GP^{(\beta)}$ - and $GQ^{(\beta)}$ -positive.* The involution $\omega$ from (2.4) extends to a continuous linear map $\mathfrak{m}\mathsf{Sym}\to\mathfrak{m}\mathsf{Sym}$ , which restricts to an algebra morphism (but not an automorphism) $\Gamma^{(\beta)}\to\mathfrak{m}\mathsf{Sym}$ satisfying [23, Thm. 6.2] $$ \omega(G^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=G^{(\beta)}_{\lambda^{\top}/\hskip-2.84526pt/\mu^{\top}}(\tfrac{x_{1}}{1-\beta x_{1}},\tfrac{x_{2}}{1-\beta x_{2}},\tfrac{x_{3}}{1-\beta x_{3}},\dots) $$ where $\lambda^{\top}$ is the transpose of a partition $\lambda$ . In $\Gamma_{P}^{(\beta)}$ and $\Gamma_{Q}^{(\beta)}$ we have the simpler formulas $$ \omega(GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(\tfrac{x_{1}}{1-\beta x_{1}},\tfrac{x_{2}}{1-\beta x_{2}},\dots)\quad\text{and}\quad\omega(GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(\tfrac{x_{1}}{1-\beta x_{1}},\tfrac{x_{2}}{1-\beta x_{2}},\dots) $$ which do not involve any change in the strict partition indices [13, Prop. 5.6 and Cor. 6.6]. Let $\Omega^{(\beta)}:\Gamma^{(\beta)}\to\Gamma^{(\beta)}$ be the linear map with $$ \Omega^{(\beta)}(G^{(\beta)}_{\lambda})=G^{(\beta)}_{\lambda^{\top}}\quad\text{for all partitions $\lambda$}. $$ Then let $\Phi^{(\beta)}$ and $\Psi^{(\beta)}$ be the maps $\mathfrak{m}\mathsf{Sym}\to\mathfrak{m}\mathsf{Sym}$ with the formulas $$ \Phi^{(\beta)}(f)=f(\tfrac{x_{1}}{1+\beta x_{1}},\tfrac{x_{2}}{1+\beta x_{2}},\tfrac{x_{3}}{1+\beta x_{3}},\dots)\quad\text{and}\quad\Psi^{(\beta)}(f)=f(\tfrac{x_{1}}{1-\beta x_{1}},\tfrac{x_{2}}{1-\beta x_{2}},\tfrac{x_{3}}{1-\beta x_{3}},\dots). $$ **Lemma 3.11** *The map $\Omega^{(\beta)}$ is a bialgebra involution of $\Gamma^{(\beta)}$ with $\Omega^{(\beta)}=\Phi^{(\beta)}\circ\omega=\omega\circ\Psi^{(\beta)}$ and $$ \Omega^{(\beta)}(G^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu})=G^{(\beta)}_{\lambda^{\top}/\hskip-2.84526pt/\mu^{\top}},\quad\Omega^{(\beta)}(GP^{(\beta)}_{\nu/\hskip-2.84526pt/\kappa})=GP^{(\beta)}_{\nu/\hskip-2.84526pt/\kappa},\quad\text{and}\quad\Omega^{(\beta)}(GQ^{(\beta)}_{\nu/\hskip-2.84526pt/\kappa})=GQ^{(\beta)}_{\nu/\hskip-2.84526pt/\kappa} $$ for all partitions $\lambda$ , $\mu$ and all strict partitions $\nu$ , $\kappa$ .* * Proof* The identities (3.9) and (3.10) imply that $\Omega^{(\beta)}=\Phi^{(\beta)}\circ\omega$ is a bialgebra morphism, and that the last three formulas hold. Since $\omega$ is an involution we have $(\Phi^{(\beta)}\circ\omega)\circ(\omega\circ\Psi^{(\beta)})=(\omega\circ\Psi^{(\beta)})\circ(\Phi^{(\beta)}\circ\omega)=1$ , so as $\Omega^{(\beta)}$ is also an involution it must hold that $\Omega^{(\beta)}=\Phi^{(\beta)}\circ\omega=\omega\circ\Psi^{(\beta)}$ . ∎ The multiplication formulas in [12] (restated as [24, Eq. (5)]) imply that $$ G^{(\beta)}_{1}G^{(\beta)}_{\mu}=\sum_{\lambda}\beta^{|\lambda|-|\mu|}G^{(\beta)}_{\lambda} $$ where the sum is over all partitions $\lambda$ with $\mu\subsetneq\lambda$ such that $\mathsf{D}_{\lambda}\setminus\mathsf{D}_{\mu}$ is a rook strip in the sense of containing at most one box in each row and column. For example, $$ G^{(\beta)}_{1}G^{(\beta)}_{(3,1)}=G^{(\beta)}_{(4,1)}+G^{(\beta)}_{(3,2)}+G^{(\beta)}_{(3,1,1)}+\beta G^{(\beta)}_{(4,2)}+\beta G^{(\beta)}_{(4,1,1)}+\beta G^{(\beta)}_{(3,2,1)}+\beta^{2}G^{(\beta)}_{(4,2,1)}. $$ Similarly, if $\mu$ is a strict partition then [3, Cor. 4.8] implies that $$ GP^{(\beta)}_{1}GP^{(\beta)}_{\mu}=\sum_{\lambda}\beta^{|\lambda|-|\mu|}GP^{(\beta)}_{\lambda} $$ where the sum is over all strict partitions $\lambda\supsetneq\mu$ such that $\mathsf{SD}_{\lambda}\setminus\mathsf{SD}_{\mu}$ is a rook strip. For example, $$ GP^{(\beta)}_{1}GP^{(\beta)}_{(3,1)}=GP^{(\beta)}_{(4,1)}+GP^{(\beta)}_{(3,2)}+\beta GP^{(\beta)}_{(4,2)}. $$ We also have $$ (GP^{(\beta)}_{1})^{2}=GP^{(\beta)}_{2}\qquad\text{and}\qquad\left(1+\beta GP^{(\beta)}_{1}\right)^{2}=1+GQ^{(\beta)}_{1}. $$ The rule to expand the product of $GQ^{(\beta)}_{1}$ and $GQ^{(\beta)}_{\lambda}$ in the $GQ^{(\beta)}$ -basis is more complicated [3, Cor. 5.6]. However, the formulas in [3, 12] imply that $\Gamma^{(\beta)}$ , $\Gamma_{P}^{(\beta)}$ , and $\Gamma_{Q}^{(\beta)}$ are generated as $\mathbb{R}$ -algebras by the respective sets of elements $\{G^{(\beta)}_{n}:n\in\mathbb{Z}_{>0}\}$ , $\{GP^{(\beta)}_{n}:n\in\mathbb{Z}_{>0}\}$ , and $\{GQ^{(\beta)}_{n}:n\in\mathbb{Z}_{>0}\}$ . ### 3.3 Unsigned specializations We now consider the bialgebras $\Gamma_{Q}\subsetneq\Gamma_{P}\subsetneq\Gamma$ obtained from $\Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\subsetneq\Gamma^{(\beta)}$ by setting $\beta=1$ . This section contains the proof of Theorem 1.2 from the introduction. Before presenting this, we require some preliminaries on the classification of the $G$ -positive specializations of $\Gamma$ from [24]. Let $a=(a_{1}\geq a_{2}\geq a_{3}\geq\dots\geq 0)$ be a sequence of nonnegative real numbers with finite sum. Yeliussizov [24, Lem. 4.11] has shown that the map $\phi_{a}:\mathsf{Sym}\to\mathbb{R}$ extends to a $G$ -positive specialization of $\Gamma$ in the sense that if we write $G_{\lambda}=\sum_{\mu\supseteq\lambda}r_{\mu/\lambda}s_{\mu}$ for integers $r_{\mu/\lambda}\in\mathbb{Z}_{\geq 0}$ (which is possible by results in [12]), then the sum $\sum_{\mu\supseteq\lambda}r_{\mu/\lambda}\phi_{a}(s_{\mu})$ always converges. The map $\pi_{\gamma}:\mathsf{Sym}\to\mathbb{R}$ similarly extends to a $G$ -positive specialization of $\Gamma$ for any $\gamma\in\mathbb{R}_{\geq 0}$ , while the composition $\varepsilon_{a}=\phi_{a}\circ\omega$ extends to a well-defined algebra morphism $\Gamma\to\mathbb{R}$ (which is then another $G$ -positive specialization) if and only if $\max(a)<1$ [24, Lem. 4.11]. Following [24], we write these respective extensions as $$ \widehat{\phi}_{a}:\Gamma\to\mathbb{R},\quad\widehat{\pi}_{\gamma}:\Gamma\to\mathbb{R},\quad\text{and}\quad\widehat{\varepsilon}_{a}:\Gamma\to\mathbb{R} $$ We now present an alternate form of Theorem 1.1 from the introduction. **Theorem 3.12 ([24])** *The $G$ -positive specializations of $\Gamma$ are the maps $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ where $\gamma\in\mathbb{R}_{\geq 0}$ and $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ and $b=(1>b_{1}\geq b_{2}\geq\dots\geq 0)$ are real sequences with $$ \textstyle\prod_{n=1}^{\infty}\frac{1+a_{n}}{1-b_{n}}\in\mathbb{R}. $$ The representation of $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ is unique and it holds that $\textstyle\rho(G_{1})=-1+e^{\gamma}\prod_{n=1}^{\infty}\frac{1+a_{n}}{1-b_{n}}.$* **Remark 3.13** *The preceding result is equivalent to Theorem 1.1 since by [24, Lem. 4.7] we have $$ \textstyle\sum_{n=0}^{\infty}\left(G^{(\beta)}_{n}+\beta G^{(\beta)}_{n+1}\right)z^{n}=\left(1+\beta G^{(\beta)}_{1}\right)H(z). $$ Using this identity and (2.8), (2.9), and (2.10), one checks that if $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ then (1.3) holds with $1+\rho(G_{1})=C<\infty$ as in (1.2). Conversely, if these identities hold for a specialization $\rho:\Gamma\to\mathbb{R}$ then we must have $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ since the coefficients of the power series $\sum_{n\geq 0}\rho(G_{n}+G_{n+1})z^{n}$ determine the values of $\rho$ on the set of algebra generators $\{G_{n}:n\in\mathbb{Z}_{>0}\}\subset\Gamma$ . The uniqueness of the parameters $a$ , $b$ , and $\gamma$ follows by considering the zeros and poles of (1.3). Finally, we have $e^{\gamma}\prod_{n=1}^{\infty}\frac{1+a_{n}}{1-b_{n}}\in[1,\infty)$ if and only if $\prod_{n=1}^{\infty}\frac{1+a_{n}}{1-b_{n}}$ converges to a finite value.* We wish to use Yeliussizov’s results to find a similar classification all $GP$ - and $GQ$ -positive specializations of $\Gamma_{P}$ and $\Gamma_{Q}$ . This requires some nontrivial lemmas. First, let $\lambda$ be a partition with exactly $n$ nonzero parts. Write $\delta=(n,\dots,3,2,1)$ for the $n$ -part staircase partition so that $\lambda+\delta=(\lambda_{1}+n,\lambda_{2}+n-1,\dots,\lambda_{n}+1)$ is a strict partition. Then there exists a unique linear map $\Theta^{(\beta)}:\Gamma^{(\beta)}\to\Gamma_{Q}^{(\beta)}$ with the formula $$ \Theta^{(\beta)}(G^{(\beta)}_{\lambda})=GP^{(\beta)}_{(\lambda+\delta)/\delta}=GQ^{(\beta)}_{(\lambda+\delta)/\delta}. $$ In fact, this map is a bialgebra morphism [13, Cor. 5.16] and it is evident that $$ \Theta^{(\beta)}(G^{(\beta)}_{n})=GQ^{(\beta)}_{n}\quad\text{for all }n\in\mathbb{Z}_{\geq 0}. $$ The following less obvious lemmas rely on some theorems recently proved in [17]. **Lemma 3.14** *If $\mu\subseteq\lambda$ are any partitions then $$ \Theta^{(\beta)}(G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu})\in\mathbb{Z}_{\geq 0}[\beta]\textnormal{-span}\left\{GP^{(\beta)}_{\nu}:\nu\text{ strict}\right\}\cap\mathbb{Z}_{\geq 0}[\beta]\textnormal{-span}\left\{GQ^{(\beta)}_{\nu}:\nu\text{ strict}\right\}. $$* * Proof* As each $G^{(\beta)}_{\lambda{/\penalty 50\hskip-2.84526pt/\penalty 50}\mu}$ is a $\mathbb{Z}_{\geq 0}[\beta]$ -linear combination of $G^{(\beta)}_{\nu}$ ’s [2], we may assume $\mu=\emptyset$ . Then it suffices to show that $GP^{(\beta)}_{(\lambda+\delta)/\delta}=GQ^{(\beta)}_{(\lambda+\delta)/\delta}$ is both a $\mathbb{Z}_{\geq 0}[\beta]$ -linear combination of $GP^{(\beta)}$ -functions and a $\mathbb{Z}_{\geq 0}[\beta]$ -linear combination of $GQ^{(\beta)}$ -functions. This is precisely [17, Cor. 4.44]. ∎ **Lemma 3.15** *We have $\Theta^{(\beta)}\circ\Omega^{(\beta)}=\Theta^{(\beta)}$ as maps $\Gamma^{(\beta)}\to\Gamma_{Q}^{(\beta)}$ .* * Proof* Since $\Theta^{(\beta)}$ and $\Omega^{(\beta)}$ are bialgebra morphism, it suffices to check this identity on the generators $G^{(\beta)}_{n}$ for $\Gamma^{(\beta)}$ . Fix $n\in\mathbb{Z}_{>0}$ . Thus, we just need to show that $$ GQ^{(\beta)}_{(n+1,\dots,4,3,2)/(n,\dots,3,2,1)}=\Theta^{(\beta)}(G^{(\beta)}_{(1^{n})})=\Theta^{(\beta)}\circ\Omega^{(\beta)}(G^{(\beta)}_{n})\quad\text{is equal to}\quad\Theta^{(\beta)}(G^{(\beta)}_{n})=GQ^{(\beta)}_{n}. $$ Here is a weight-preserving bijection $\mathsf{ShSVT}_{Q}((n))\to\mathsf{ShSVT}_{Q}((n+1,\dots,4,3,2)/(n,\dots,3,2,1))$ that realizes this identity. Choose $T\in\mathsf{ShSVT}_{Q}((n))$ . For each $i\in\mathbb{Z}_{>0}$ , let $i^{\prime}=i-\frac{1}{2}$ and consider the boxes of $T$ containing $i$ or $i^{\prime}$ (or both). These boxes must be adjacent and of the form when we ignore all entries except $i$ or $i^{\prime}$ . 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17.07182pt\hrule width=17.81169pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to17.44176pt{\vss\hbox to17.07182pt{\hss$i^{\prime}i$\hss}\vss}\kern-17.44176pt\vrule height=17.44176pt,width=0.36993pt\kern 17.07182pt\vrule height=17.44176pt,width=0.36993pt}\kern-0.18497pt\kern-17.44176pt\hrule width=17.81169pt,height=0.36993pt\kern 17.07182pt\hrule width=17.81169pt,height=0.36993pt}\crcr}}\kern 36.02338pt}}$}}. $$ After performing this operation for all $i\in\mathbb{Z}_{>0}$ , transpose the boxes of $T$ to obtain a tableau $U$ with $n$ boxes arranged in one column. The desired bijection is then $T\mapsto U$ . ∎ Our final lemma in this section is a shifted version of (3.17). Here, we define $$ \overline{x}=\frac{-x}{1+x}\quad\text{for any parameter $x$}. $$ **Lemma 3.16 ([8])** *It holds that $$ \sum_{n=0}^{\infty}\left(GQ^{(\beta)}_{n}+\beta GQ^{(\beta)}_{n+1}\right)z^{n}=\left(1+\beta G^{(\beta)}_{1}\right)E(z+\beta)H(z). $$* * Proof* After unpacking the notation in [8, §10], this identity follows from [8, Cor. 10.10]. ∎ The preceding lemma can also be shown by a more direct combinatorial argument. In any event, this brings us to our shifted version of Theorem 3.12. **Theorem 3.17** *The $GP$ -positive (respectively, $GQ$ -positive) specializations of $\Gamma_{P}$ (respectively, $\Gamma_{Q}$ ) are the restrictions of the maps $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\pi}_{\gamma}$ where $\gamma\in\mathbb{R}_{\geq 0}$ and where $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ is a real sequence with $\prod_{n=1}^{\infty}(1+a_{n})\in\mathbb{R}$ . For such maps one has $$ \sum_{n\geq 0}\rho(GQ_{n}+GQ_{n+1})z^{n}=D^{2}e^{2\gamma z}\prod_{n=1}^{\infty}\frac{1-\overline{a_{n}}z}{1-a_{n}z} $$ where $D=1+\rho(G_{1})=e^{\gamma}\prod_{n=1}^{\infty}(1+a_{n})$ and $D^{2}=1+\rho(GQ_{1})$ .* * Proof* In view of Proposition 3.10, any such map $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\pi}_{\gamma}$ restricts to $GP$ - and $GQ$ -positive specializations of $\Gamma_{P}$ of $\Gamma_{Q}$ , and we have $\rho(GP_{1})=\rho(G_{1})=-1+e^{\gamma}\prod_{n=1}^{\infty}(1+a_{n})$ by Theorem 3.12. Then by using Lemma 3.16 with (2.8), (2.9), and (2.10), one can check that (3.21) holds. Now suppose $\rho$ is a $GP$ -positive specialization of $\Gamma_{P}$ or a $GQ$ -positive specialization of $\Gamma_{Q}$ . Let $\Theta=\Theta^{(1)}$ and $\Omega=\Omega^{(1)}$ . Then Lemmas 3.14 and 3.15 imply that $\rho\circ\Theta=\rho\circ\Theta\circ\Omega$ is a $\overline{G}$ -positive specialization of $\overline{\Gamma}$ , so $\rho\circ\Theta=\widehat{\phi}_{a}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ for unique parameters as in Theorem 3.12. As $\Omega$ is a bialgebra morphism, we have $$ (\widehat{\phi}_{a}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma})\circ\Omega=(\widehat{\phi}_{a}\circ\Omega)\sqcup(\widehat{\varepsilon}_{b}\circ\Omega)\sqcup(\widehat{\pi}_{\gamma}\circ\Omega). $$ The formulas (2.5), (2.6), and $\Omega=\Phi^{(1)}\circ\omega=\omega\circ\Psi^{(1)}$ in Lemma 3.11 imply that $$ \widehat{\phi}_{a}\circ\Omega=\widehat{\varepsilon}_{a^{\prime}},\quad\widehat{\varepsilon}_{b}\circ\Omega=\widehat{\phi}_{b^{\prime}},\quad\text{and}\quad\widehat{\pi}_{\gamma}\circ\Omega=\widehat{\pi}_{\gamma} $$ for $a^{\prime}=(\frac{a_{1}}{1+a_{1}},\frac{a_{2}}{1+a_{2}},\dots)$ and $b^{\prime}=(\frac{b_{1}}{1-b_{1}},\frac{b_{2}}{1-b_{2}},\dots)$ . Since $\widehat{\phi}_{a}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}=(\widehat{\phi}_{a}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma})\circ\Omega$ , the uniqueness of our parameters means that we must have $a=b^{\prime}$ and $b=a^{\prime}$ . Thus, Theorems 1.1 and 3.12 imply that $$ \sum_{n\geq 0}\rho(GQ_{n}+GQ_{n+1})z^{n}=\sum_{n\geq 0}\rho\circ\Theta(G_{n}+G_{n+1})z^{n}=(1+\rho(GQ_{1}))e^{\gamma z}\prod_{n=1}^{\infty}\frac{1-\overline{a_{n}}z}{1-a_{n}z} $$ where we have $$ 1+\rho(GQ_{1})=1+\rho\circ\Theta(G^{(\beta)}_{1})=e^{\gamma}\prod_{n=1}^{\infty}\frac{1+a_{n}}{1+\overline{a_{n}}}=e^{\gamma}\prod_{n=1}^{\infty}(1+a_{n})^{2}, $$ so the infinite product $\prod_{n=1}^{\infty}(1+a_{n})$ must converge. Again using Lemma 3.16 with (2.8), (2.9), and (2.10), one checks that the right side of (3.22) is equal to $\sum_{n\geq 0}(\widehat{\phi}_{a}\sqcup\widehat{\pi}_{\gamma/2})(GQ_{n}+GQ_{n+1})z^{n}.$ Thus $\rho$ must coincide with the restriction of $\widehat{\phi}_{\alpha}\sqcup\widehat{\pi}_{\gamma/2}$ since the coefficients of $$ \sum_{n\geq 0}\rho(GQ_{n}+GQ_{n+1})z^{n}=(1+\rho(GP_{1}))^{2}+\sum_{n\geq 1}\rho(2GP_{n}+3GP_{n+1}+GP_{n+2})z^{n} $$ determine the values of $\rho$ on the relevant set of algebra generators $\{GP_{n}\}\subset\Gamma_{P}$ or $\{GQ_{n}\}\subset\Gamma_{Q}$ (as $1+\rho(GP_{1})$ is positive and $GP_{2}=(GP_{1})^{2}$ ). Replacing $\gamma$ by $2\gamma$ gives the desired form of $\rho$ . ∎ * Proof of Theorem1.2* In the proof of Theorem 3.17, we showed that the $GP$ -positive specializations of $\Gamma_{P}$ and the $GQ$ -positive specializations of $\Gamma_{Q}$ are obtained by restricting the same $G$ -positive specializations $\rho$ of $\Gamma$ , and that (3.21) holds for each such specialization. We also observed in our argument that if (3.21) holds then $\rho$ coincides with the restriction of $\widehat{\phi}_{a}\sqcup\widehat{\pi}_{\gamma}$ for appropriate parameters $a$ and $\gamma$ , so $\rho$ restricts to $GP$ - and $GQ$ -positive specializations by Proposition 3.10 and Theorem 3.12. This confirms the equivalence of (a), (b), and (c) in Theorem 1.2. ∎ ### 3.4 Single-variable formulas The results in this section, which are needed to classify the $\overline{GP}$ - and $\overline{GQ}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ , are $GP^{(\beta)}$ - and $GQ^{(\beta)}$ -analogues of some $G^{(\beta)}$ -function formulas in [23]. Let $\mu\subseteq\lambda$ be strict partitions such that the shifted skew shape $\mathsf{SD}_{\lambda/\mu}:=\mathsf{SD}_{\lambda}\setminus\mathsf{SD}_{\mu}$ is a border strip in the sense that $(i,j)\in\mathsf{SD}_{\lambda/\mu}$ implies that $(i+1,j+1)\notin\mathsf{SD}_{\lambda/\mu}$ . Define $$ \mathsf{Inn}(\lambda,\mu)=\{(i,j)\in\mathsf{SD}_{\mu}:(i+1,j)\notin\mathsf{SD}_{\mu},\ (i,j+1)\notin\mathsf{SD}_{\mu},\ (i+1,j+1)\notin\mathsf{SD}_{\lambda}\}. $$ Then let | | $\displaystyle\mathsf{Free}_{Q}(\lambda,\mu)$ | $\displaystyle=\{(i,j)\in\mathsf{SD}_{\lambda/\mu}:(i+1,j),(i,j-1)\notin\mathsf{SD}_{\lambda/\mu}\},$ | | | --- | --- | --- | --- | along with | | $\displaystyle\mathsf{Gap}_{Q}(\lambda,\mu)$ | $\displaystyle=\{(i,j)\in\mathsf{Inn}(\lambda,\mu):(i+1,j),(i,j+1)\in\mathsf{SD}_{\lambda/\mu}\},$ | | | --- | --- | --- | --- | Finally let | | $\displaystyle\mathsf{Adj}_{Q}(\lambda,\mu)$ | $\displaystyle=\{(i,j)\in\mathsf{Inn}(\lambda,\mu):(i+1,j)\in\mathsf{SD}_{\lambda/\mu}\text{ or }(i,j+1)\in\mathsf{SD}_{\lambda/\mu}\}-\mathsf{Gap}_{Q}(\lambda,\mu),$ | | | --- | --- | --- | --- | When $f\in\mathbb{R}\llbracket\mathbf{x}\rrbracket$ is a power series and $a_{1},a_{2},\dots,a_{k}$ is a finite list of parameters, we let $$ f(a_{1},a_{2},\dots,a_{k})=f(a_{1},a_{2},\dots,a_{k},0,0,0,\dots) $$ denote the result of substituting $x_{i}\mapsto a_{i}$ for $1\leq i\leq k$ and $x_{i}\mapsto 0$ for $i>k$ . **Proposition 3.18** *Let $\lambda$ and $\mu$ be strict partitions. If $\mu\not\subseteq\lambda$ or if $\mathsf{SD}_{\lambda/\mu}$ is not a border strip, then the single-variable power series $GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=0$ are both zero. Otherwise, we have | | $\displaystyle GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)$ | $\displaystyle=2^{a_{P}}(2+\beta x)^{b_{P}}(1+\beta x)^{c_{P}}x^{|\lambda|-|\mu|}\quad\text{and}\quad$ | | | --- | --- | --- | --- | where for either type $K\in\{P,Q\}$ we define | | $\displaystyle a_{K}=a_{K}(\lambda,\mu)$ | $\displaystyle=|\mathsf{Gap}_{K}(\lambda,\mu)|,$ | | | --- | --- | --- | --- |* Before giving a proof of this result, we note a corollary and present some examples. **Corollary 3.19** *If $\lambda$ is a strict partition with more than one part then $GP^{(\beta)}_{\lambda}(x)=GQ^{(\beta)}_{\lambda}(x)=0$ . Otherwise, if $n\in\mathbb{Z}_{>0}$ then $GP^{(\beta)}_{n}(x)=x^{n}$ and $GQ^{(\beta)}_{n}(x)=(2+\beta x)x^{n}$ .* **Example 3.20** *Suppose $\mu=(2)$ and $\lambda=(5)$ . Then $$ \mathsf{SD}_{\lambda/\mu}={\hbox{$\vbox{\hbox{\vtop{\halign{&\opttoksa@YT={\font@YT}\getcolor@YT{\save@YT{\opttoksb@YT}}\nil@YT\getcolor@YT{\startbox@@YT\the\opttoksa@YT\the\opttoksb@YT}#\endbox@YT\cr\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$\ $\hss}\vss}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$\ $\hss}\vss}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$\ $\hss}\vss}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}\crcr}}\kern 24.64157pt}}$}} $$ so we have | | $\displaystyle\mathsf{SD}_{\lambda/\mu}$ | $\displaystyle=\{(1,3),(1,4),(1,5)\},$ | | | --- | --- | --- | --- | along with $a_{P}=a_{Q}=0$ , $b_{P}=b_{Q}=1$ , $c_{P}=c_{P}=1$ , and $|\lambda/\mu|=3$ . Hence $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=(2+\beta x)(1+\beta x)x^{3}. $$* **Example 3.21** *Suppose $\mu=(4,2)$ and $\lambda=(4,3)$ . Then $$ \mathsf{SD}_{\lambda/\mu}={\hbox{$\vbox{\hbox{\vtop{\halign{&\opttoksa@YT={\font@YT}\getcolor@YT{\save@YT{\opttoksb@YT}}\nil@YT\getcolor@YT{\startbox@@YT\the\opttoksa@YT\the\opttoksb@YT}#\endbox@YT\cr\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}\cr\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}\kern-11.75085pt\vrule width=0.0pt,height=11.75085pt\kern 0.36993pt\kern 11.38092pt\vrule width=0.0pt,height=11.75085pt\kern 0.36993pt}\kern-0.18497pt\kern-11.75085pt\kern 0.36993pt\kern 11.38092pt\kern 0.36993pt}\nullfont&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\hbox{\pagecolor{lightgray}\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$$\hss}\vss}}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}&\lower 0.36993pt\vbox{\kern 0.18497pt\hbox{\kern 0.36993pt\vbox to11.75085pt{\vss\hbox to11.38092pt{\hss$\ $\hss}\vss}\kern-11.75085pt\vrule height=11.75085pt,width=0.36993pt\kern 11.38092pt\vrule height=11.75085pt,width=0.36993pt}\kern-0.18497pt\kern-11.75085pt\hrule width=12.12079pt,height=0.36993pt\kern 11.38092pt\hrule width=12.12079pt,height=0.36993pt}\crcr}}\kern 24.64157pt}}$}} $$ so we have | | $\displaystyle\mathsf{SD}_{\lambda/\mu}=\mathsf{Free}_{P}(\lambda,\mu)=\mathsf{Free}_{Q}(\lambda,\mu)$ | $\displaystyle=\{(2,4)\},$ | | | --- | --- | --- | --- | so $a_{P}=a_{Q}=0$ , $b_{P}=b_{Q}=1$ , $c_{P}=c_{Q}=2$ , and $|\lambda/\mu|=1$ . Hence $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)=(2+\beta x)(1+\beta x)^{2}x. $$* * Proof of Proposition3.18* We have $GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x_{1})=\sum_{T}\beta^{|T|-|\lambda/\mu|}x^{T}$ where $T$ ranges over all set-valued shifted tableaux of shape $\lambda/\hskip-2.84526pt/\mu$ whose entries are all subsets of $\{1^{\prime},1\}$ where $1^{\prime}=\frac{1}{2}$ . If $\mathsf{SD}_{\lambda/\mu}$ is not a border strip then there are no such $T$ . Assume $\mathsf{SD}_{\lambda/\mu}$ is a border strip and $T$ is one of the desired tableaux. The nonempty boxes of $T$ consist of $\mathsf{SD}_{\lambda/\mu}$ plus arbitrary subsets $$ A\subseteq\mathsf{Gap}_{Q}(\lambda,\mu),\ \ B\subseteq\mathsf{Adj}_{Q}(\lambda,\mu),\ \ \text{and}\ \ C\subseteq\mathsf{Diff}_{Q}(\lambda,\mu):=\mathsf{Inn}(\lambda,\mu)-\mathsf{Gap}_{Q}(\lambda,\mu)-\mathsf{Adj}_{Q}(\lambda,\mu). $$ Together, this set of boxes is itself a border strip that is a union of zero or more connected ribbons. Define the first box of a connected ribbon to be the one with smallest column index and largest row index. For a fixed choice of $A$ , $B$ , and $C$ , the desired tableaux $T$ are constructed as follows: every box that is not the first in its ribbon must contain just $1^{\prime}$ or just $1$ , and the choice is uniquely determined, while the first boxes in each connected ribbon may independently contain $1^{\prime}$ , $1$ , or both $1^{\prime}$ and $1$ . For all such $T$ , the number of filled boxes is $|\lambda/\mu|+|A|+|B|+|C|$ while the number of connected ribbons is $|\mathsf{Free}_{Q}(\lambda,\mu)|-|A|+|C|$ . We deduce from these observations that | | $\displaystyle GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)$ | $\displaystyle=\sum_{\begin{subarray}{c}A\subseteq\mathsf{Gap}_{Q}(\lambda,\mu)\\ B\subseteq\mathsf{Adj}_{Q}(\lambda,\mu)\\ C\subseteq\mathsf{Diff}_{Q}(\lambda,\mu)\end{subarray}}\beta^{|A|+|B|+|C|}x^{|\lambda/\mu|+|A|+|B|+|C|}(2+\beta x)^{|\mathsf{Free}_{Q}(\lambda,\mu)|-|A|+|C|}$ | | | --- | --- | --- | --- | Via the binomial theorem, this expression becomes $$ (2+\beta x)^{b_{Q}}x^{|\lambda/\mu|}(2+2\beta x)^{|\mathsf{Gap}_{Q}(\lambda,\mu)|}(1+\beta x)^{|\mathsf{Adj}_{Q}(\lambda,\mu)|+2|\mathsf{Diff}_{Q}(\lambda,\mu)|}=2^{a_{Q}}(2+\beta x)^{b_{Q}}(1+\beta x)^{c_{Q}}x^{|\lambda/\mu|} $$ as desired. The argument to derive the formula for $GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(x)$ is similar, with a few adjustments to incorporate the extra requirement that $1^{\prime}\notin T_{ij}$ if $i=j$ . ∎ Given a strict partition $\lambda=(\lambda_{1}>\lambda_{2}>\dots\geq 0)$ , let $\tilde{\lambda}=(\lambda_{2}>\lambda_{3}>\dots\geq 0)$ . **Corollary 3.22** *If $\lambda$ and $\mu$ are strict partitions then $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=\begin{cases}(-\tfrac{1}{\beta})^{\lambda_{1}}&\text{if }\mu=\tilde{\lambda}\\ 0&\text{otherwise}.\end{cases} $$* * Proof* Proposition 3.18 implies that $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=0 $$ unless $\mu\subseteq\lambda$ and $\mathsf{SD}_{\lambda/\mu}$ is a border strip, in which case $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=2^{a_{P}}\cdot 0^{c_{P}}\cdot(-\beta)^{|\mu|-|\lambda|}\quad\text{and}\quad GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta})=2^{a_{Q}}\cdot 0^{c_{Q}}\cdot(-\beta)^{|\mu|-|\lambda|}. $$ As $\mathsf{Gap}_{K}(\lambda,\mu)$ and $\mathsf{Adj}_{K}(\lambda,\mu)$ are disjoint subsets of $\mathsf{Inn}(\lambda,\mu)$ , both $c_{P}$ and $c_{Q}$ are positive unless $\mathsf{Inn}(\lambda,\mu)=\varnothing$ , and then $a_{P}=c_{Q}=a_{P}=c_{Q}=0$ . But $\mathsf{SD}_{\lambda/\mu}$ is a border strip with $\mathsf{Inn}(\lambda,\mu)=\varnothing$ precisely when $\mu=\tilde{\lambda}$ and then $|\mu|-|\lambda|=-\lambda_{1}$ . ∎ We are led to a shifted analogue of [24, Lem. 6.7]. Recall that $GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\nu}=GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\nu}=0$ if $\nu\not\subseteq\lambda$ . **Corollary 3.23** *If $\mu\subseteq\lambda$ are strict partitions with $\lambda\neq\emptyset$ then $$ GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta},x_{1},x_{2},x_{3},\dots)=(-\tfrac{1}{\beta})^{\lambda_{1}}GP^{(\beta)}_{\tilde{\lambda}/\hskip-2.84526pt/\mu}\quad\text{and}\quad GQ^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta},x_{1},x_{2},x_{3},\dots)=(-\tfrac{1}{\beta})^{\lambda_{1}}GQ^{(\beta)}_{\tilde{\lambda}/\hskip-2.84526pt/\mu}. $$* * Proof* Notice that $GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\mu}(-\tfrac{1}{\beta},x_{1},x_{2},x_{3},\dots)=\sum_{\mu\subseteq\kappa\subseteq\lambda}GP^{(\beta)}_{\lambda/\hskip-2.84526pt/\kappa}(-\tfrac{1}{\beta})GP^{(\beta)}_{\kappa/\hskip-2.84526pt/\mu}$ and then substitute the formula in Corollary 3.22. The other formula is derived in a similar way. ∎ The case when $\mu=\lambda=\emptyset$ is excluded above since $GP^{(\beta)}_{\emptyset/\hskip-2.84526pt/\emptyset}=GQ^{(\beta)}_{\emptyset/\hskip-2.84526pt/\emptyset}=1$ . For any partition $\lambda$ let $\ell(\lambda)=|\{i\in\mathbb{Z}_{>0}:\lambda_{i}>0\}|$ denote its number of nonzero parts. Repeatedly applying Corollary 3.23 recovers a result of Nobukawa and Shimazaki [19, Cor. 4.1]. **Corollary 3.24 ([19])** *If $\emptyset\neq\mu\subseteq\lambda$ are strict partitions then | | $\displaystyle GP^{(\beta)}_{\lambda}(-\tfrac{1}{\beta},-\tfrac{1}{\beta},\dots,-\tfrac{1}{\beta})$ | $\displaystyle=GQ^{(\beta)}_{\lambda}(-\tfrac{1}{\beta},-\tfrac{1}{\beta},\dots,-\tfrac{1}{\beta})=(-\tfrac{1}{\beta})^{|\lambda|}\quad\text{and}\quad$ | | | --- | --- | --- | --- | whenever the number of variable set to $-\frac{1}{\beta}$ respectively exceeds $\ell(\lambda)$ and $\ell(\lambda)-\ell(\mu)$ .* ### 3.5 Signed specializations Finally, we consider the instances $\overline{\Gamma}_{Q}\subsetneq\overline{\Gamma}_{P}\subsetneq\overline{\Gamma}$ of the bialgebras $\Gamma_{Q}^{(\beta)}\subsetneq\Gamma_{P}^{(\beta)}\subsetneq\Gamma^{(\beta)}$ with $\beta=-1$ . Our goal in this section is to classify the $\overline{GP}$ - and $\overline{GQ}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ . We do this after reviewing Yeliussizov’s classifications of the $\overline{G}$ -positive specializations of $\overline{\Gamma}$ . **Lemma 3.25** *If $f$ , $g$ , and $h$ are $\overline{G}$ -, $\overline{GP}$ -, are $\overline{GQ}$ -positive specializations of $\overline{\Gamma}$ , $\overline{\Gamma}_{P}$ , and $\overline{\Gamma}_{Q}$ , then $$ \{f(\overline{G}_{n}):n=0,1,2,\dots\},\quad\{g(\overline{GP}_{n}):n=0,1,2,\dots\},\quad\text{and}\quad\{h(\overline{GQ}_{n}):n=0,1,2,\dots\} $$ are all weakly decreasing sequences of real numbers in the interval $[0,1]$ .* * Proof* The claim about $f$ is [24, Lem. 6.3]. The claims about $g$ and $h$ hold since we have $$ \overline{GP}_{n}-\overline{GP}_{n+1}=\tfrac{1}{2}\left(\overline{GQ}_{n}-\overline{GP}_{n+1}\right)=\tfrac{1}{2}\overline{GP}_{(n+1){/\penalty 50\hskip-2.84526pt/\penalty 50}(1)}\quad\text{and}\quad\overline{GQ}_{n}-\overline{GQ}_{n+1}=\overline{GQ}_{(n+1){/\penalty 50\hskip-2.84526pt/\penalty 50}(1)} \tag{1} $$ for all $n\in\mathbb{Z}_{\geq 0}$ by Example 3.6 and (3.5), along with $g(\overline{GP}_{0})=g(1)=h(\overline{GQ}_{0})=h(1)=1$ . ∎ Let $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ be a sequence of nonnegative real numbers with finite sum. It follows from [24, Lem. 6.5] that the map $\phi_{a}:\mathsf{Sym}\to\mathbb{R}$ extends to a specialization of $\overline{\Gamma}$ if and only if $\max(a)\leq 1$ . The same result implies that $\varepsilon_{a}:\mathsf{Sym}\to\mathbb{R}$ and $\pi_{\gamma}:\mathsf{Sym}\to\mathbb{R}$ extend to specializations of $\overline{\Gamma}$ for all choices of $a=(a_{1}\geq a_{2}\geq\dots\geq 0)$ and $\gamma\in\mathbb{R}_{\geq 0}$ . We denote these extended specializations using the same symbols as in Section 3.3, namely: $$ \widehat{\phi}_{a}:\Gamma\to\mathbb{R},\quad\widehat{\varepsilon}_{a}:\Gamma\to\mathbb{R},\quad\text{and}\quad\widehat{\pi}_{\gamma}:\Gamma\to\mathbb{R}. $$ The following combines [24, Thm. 6.4 and Prop. 6.6]. **Theorem 3.26 ([24])** *If $\delta\in[0,1)$ and $\rho$ is a specialization of $\overline{\Gamma}$ then the following are equivalent: 1. The map $\rho$ is a $\overline{G}$ -positive specialization of $\overline{\Gamma}$ with $\rho(\overline{G}_{1})=\delta$ . 1. It holds that $\rho=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ for unique sequences of real numbers $$ a=(1>a_{1}\geq a_{2}\geq\dots\geq 0),\quad b=(b_{1}\geq b_{2}\geq\dots\geq 0),\quad\text{and}\quad\gamma\geq 0 $$ satisfying $\delta=1-e^{-\gamma}\prod_{n=1}^{\infty}\frac{1-a_{n}}{1+b_{n}}$ . 1. One has $\sum_{n=0}^{\infty}\rho\left(\overline{G}_{n}-\overline{G}_{n+1}\right)z^{n}=(1-\delta)e^{\gamma z}\prod_{n=1}^{\infty}\frac{1+b_{n}z}{1-a_{n}z}$ for some parameters as in (b).* Although any $\overline{G}$ -positive specialization of $\overline{\Gamma}$ restricts to a specialization of the subalgebras $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ , it does not follow immediately that these restrictions are $GP^{(\beta)}$ - or $GQ^{(\beta)}$ -positive. **Lemma 3.27** *Let $a=(1\geq a_{1}\geq a_{2}\geq\dots\geq 0)$ and $b=(b_{1}\geq b_{2}\geq\dots\geq 0)$ be sequences of real numbers with finite sum and let $\gamma\in\mathbb{R}_{\geq 0}$ . Then $\widehat{\phi}_{a}$ , $\widehat{\varepsilon}_{b}$ , and $\widehat{\pi}_{\gamma}$ restrict to $GP^{(\beta)}$ - and $GQ^{(\beta)}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ .* * Proof* The map $\widehat{\phi}_{a}$ is the union of the one-variable specializations $\widehat{\phi}_{a_{i}}$ , which restrict to $GQ^{(\beta)}$ - and $GP^{(\beta)}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ when $a_{i}\in[0,1]$ by Proposition 3.18 with $\beta=-1$ . Hence, the same is true of $\widehat{\phi}_{a}$ . Next, by Lemma 3.11 with $\beta=-1$ we have $\widehat{\varepsilon}_{b}\circ\Omega^{(-1)}=\widehat{\phi}_{b}\circ\omega\circ\omega\circ\Psi^{(-1)}=\widehat{\phi}_{b}\circ\Psi^{(-1)}$ which is equal to $\widehat{\phi}_{c}$ for the sequence $c=(\frac{b_{1}}{1+b_{1}},\frac{b_{2}}{1+b_{2}},\dots)$ . Since $\Omega^{(-1)}$ restricts to the identity on $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ , the desired claim about $\widehat{\varepsilon}_{b}$ follows from what was already shown for $\widehat{\phi}_{a}$ . Finally, $\widehat{\pi}_{\gamma}$ restricts to $GQ^{(\beta)}$ - and $GP^{(\beta)}$ -positive specializations of $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ since by (2.5) the value of $\widehat{\pi}_{\gamma}(f)$ is the limit as $N\to\infty$ of $\widehat{\phi}_{a}(f)$ where $a=(\gamma/N,\gamma/N,\dots,\gamma/N)\in\mathbb{R}^{N}$ . ∎ The following is now clear from Theorem 3.26 and (3.5) given Remark 3.9 and Lemma 3.27. **Corollary 3.28** *A $\overline{G}$ -positive specialization $\rho$ of $\overline{\Gamma}$ restricts to a $\overline{GP}$ -positive specialization of $\overline{\Gamma}_{P}$ with $\rho(\overline{GP}_{1})=\rho(\overline{G}_{1})$ and to a $\overline{GQ}$ -positive specialization of $\overline{\Gamma}_{Q}$ with $\rho(\overline{GQ}_{1})=1-(1-\rho(\overline{G}_{1}))^{2}$ .* As an alternative to (3.20), define $$ \widetilde{x}=\frac{-x}{1-x}\quad\text{for any parameter $x$}. $$ A specialization of $\overline{\Gamma}$ , $\overline{\Gamma}_{P}$ , or $\overline{\Gamma}_{Q}$ is normalized if its value at $\overline{G}_{1}$ , $\overline{GP}_{1}$ , or $\overline{GQ}_{1}$ is equal to one, and unnormalized if this value is strictly less than one. Corollary 3.28 implies that a specialization of $\overline{\Gamma}$ is normalized if and only if its restrictions to $\overline{\Gamma}_{P}$ and $\overline{\Gamma}_{Q}$ are normalized. **Theorem 3.29** *The unnormalized $\overline{GP}$ -positive specializations of $\overline{\Gamma}_{P}$ and the unnormalized $\overline{GQ}$ -positive specializations of $\overline{\Gamma}_{Q}$ are each obtained by restricting some unnormalized $G$ -positive specialization of $\Gamma$ . If $\delta\in[0,1)$ and $\rho$ is a specialization of $\overline{\Gamma}$ then the following are equivalent: 1. The map $\rho$ restricts to a $\overline{GP}$ -positive specialization of $\overline{\Gamma}_{P}$ with $\rho(\overline{GP}_{1})=\delta$ . 1. The map $\rho$ restricts to a $\overline{GQ}$ -positive specialization of $\overline{\Gamma}_{Q}$ with $\rho(\overline{GQ}_{1})=1-(1-\delta)^{2}$ . 1. The restriction of $\rho$ to $\overline{\Gamma}_{P}$ coincides with $\widehat{\phi}_{\alpha}\sqcup\widehat{\pi}_{\gamma}$ for some real numbers $$ a=(1>a_{1}\geq a_{2}\geq\dots\geq 0)\quad\text{and}\quad\gamma\geq 0\quad\text{satisfying $\textstyle\delta=1-e^{-\gamma}\prod_{n=1}^{\infty}(1-a_{n})$.} $$ 1. One has $\displaystyle\sum_{n=0}^{\infty}\rho\left(\overline{GQ}_{n}-\overline{GQ}_{n+1}\right)z^{n}=(1-\delta)^{2}e^{2\gamma z}\prod_{n=1}^{\infty}\frac{1-\widetilde{a}_{n}z}{1-a_{n}z}$ for some parameters as in (c).* * Proof* Parts (c) and (d) are equivalent by Lemma 3.16 using the identities (2.8), (2.9), (2.10) via the argument in the proof of Theorem 3.17. In turn, part (c) implies both (a) and (b) by Theorem 3.26 and Corollary 3.28. Conversely, suppose $\rho$ is an unnormalized $\overline{GP}$ -positive specialization of $\overline{\Gamma}_{P}$ or an unnormalized $\overline{GQ}$ -positive specialization of $\overline{\Gamma}_{Q}$ . In either case, it follows from (3.6) and (3.18) with $\beta=-1$ that $\rho\circ\Theta^{(-1)}$ is a $\overline{G}$ -positive specialization of $\overline{\Gamma}$ whose value at $\overline{G}_{1}$ is $$ \rho(\overline{GQ}_{1})=\rho(2\overline{GP}_{1}-\overline{GP}_{2})=\rho(2\overline{GP}_{1}-(\overline{GP}_{1})^{2})=1-(1-\rho(\overline{GP}_{1}))^{2}<1. $$ Hence $\rho\circ\Theta^{(-1)}=\widehat{\phi}_{\alpha}\sqcup\widehat{\varepsilon}_{b}\sqcup\widehat{\pi}_{\gamma}$ for unique parameters $a$ , $b$ , and $\gamma$ as in Theorem 3.26 (b). Since $\rho\circ\Theta^{(-1)}=\rho\circ\Theta^{(-1)}\circ\Omega^{(-1)}$ by Lemma 3.11, it follows by the argument in the proof of Theorem 3.17 (just with $\beta=1$ changed to $\beta=-1$ ) that $b_{n}=-\widetilde{a}_{n}$ for all $n\in\mathbb{Z}_{>0}$ . Therefore $$ \sum_{n=0}^{\infty}\rho\left(\overline{GQ}_{n}-\overline{GQ}_{n+1}\right)z^{n}=\sum_{n=0}^{\infty}\rho\circ\Theta^{(-1)}\left(\overline{G}_{n}-\overline{G}_{n+1}\right)z^{n}=(1-\rho(\overline{GQ}_{1}))e^{\gamma z}\prod_{n=1}^{\infty}\frac{1-\widetilde{a}_{n}z}{1-a_{n}z} $$ and by Theorem 3.26 we have $\textstyle 1-\rho(\overline{GQ}_{1})=e^{-\gamma}\prod_{n=1}^{\infty}\frac{1-a_{n}}{1-\widetilde{a}_{n}}=e^{-\gamma}\prod_{n=1}^{\infty}(1-a_{n})^{2}.$ This is just part (d) with $\gamma$ replaced by $\gamma/2$ . Thus (a) $\Rightarrow$ (d) and (b) $\Rightarrow$ (d), so all four properties are equivalent. ∎ We conclude this section with a shifted version of [24, Thm. 6.8]. Let $\widehat{1}=\widehat{\phi}_{(1,0,0,0,\dots)}$ . Recall that if $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},\dots)$ then $\tilde{\lambda}=(\lambda_{2},\lambda_{3},\dots)$ . If $\lambda$ is empty or has only one nonzero part, then we interpret this notation to mean $\tilde{\lambda}=\emptyset$ . **Proposition 3.30** *Let $\varphi$ be a specialization of $\overline{\Gamma}_{P}$ (respectively, $\overline{\Gamma}_{Q}$ ). Then $\varphi^{+}:=\widehat{1}\sqcup\varphi$ satisfies $$ \varphi^{+}(\overline{GP}_{\lambda/\hskip-2.84526pt/\mu})=\varphi(\overline{GP}_{\tilde{\lambda}/\hskip-2.84526pt/\mu})\quad\text{and}\quad\varphi^{+}(\overline{GQ}_{\lambda/\hskip-2.84526pt/\mu})=\varphi(\overline{GQ}_{\tilde{\lambda}/\hskip-2.84526pt/\mu}) $$ for all strict partitions $\lambda$ and $\mu$ . Consequently, it holds that: 1. $\varphi^{+}(\overline{GP}_{n})=1$ and $\varphi^{+}(\overline{GQ}_{n})=1$ for all $n\in\mathbb{Z}_{>0}$ . 1. $\varphi^{+}$ is a $\overline{GP}$ - or $\overline{GQ}$ -positive specialization if and only if $\varphi$ has the same property.* * Proof* The identities (3.25) follow from Corollary 3.22 and (3.8). For part (a), note that if $\lambda=(n)$ and $\mu=\emptyset$ then $\tilde{\lambda}/\hskip-2.84526pt/\mu=\emptyset/\hskip-2.84526pt/\emptyset$ , and we have $\overline{GP}_{\emptyset/\hskip-2.84526pt/\emptyset}=\overline{GQ}_{\emptyset/\hskip-2.84526pt/\emptyset}=1$ and $\varphi(1)=1$ . For part (b), note that if $\varphi$ is positive then $\varphi^{+}$ is a union of positive specializations, and hence positive, while if $\varphi^{+}$ is positive then (3.25) implies that $\varphi$ is positive. ∎ Corollaries 3.23 and 3.24 specialize when $\beta=-1$ to the following results. For $f\in\overline{\Gamma}$ let $f(1^{n})$ denote the variable substitution $f(a_{1},a_{2},\dots,a_{n})$ where $a_{1}=a_{2}=\dots=a_{n}=1$ . **Corollary 3.31** *If $\lambda$ and $\mu$ are strict partitions and $n\in\mathbb{Z}_{\geq 0}$ then $$ \overline{GP}_{\lambda/\hskip-2.84526pt/\mu}(1^{n})=\overline{GQ}_{\lambda/\hskip-2.84526pt/\mu}(1^{n})=\begin{cases}1&\text{if }\mu=(\lambda_{n+1},\lambda_{n+2},\lambda_{n+3},\dots)\\ 0&\text{otherwise}.\end{cases} $$* **Corollary 3.32** *If $\lambda$ is a strict partition then $\overline{GP}_{\lambda}(1^{n})=\overline{GQ}_{\lambda}(1^{n})=\begin{cases}1&\text{if }n\geq\ell(\lambda)\\ 0&\text{if }n<\ell(\lambda).\end{cases}$* Taking the limiting case of the preceding corollary gives the following: **Corollary 3.33** *There exists a specialization $\mathbf{1}:\overline{\Gamma}\to\mathbb{R}$ with $\mathbf{1}(\overline{G}_{\lambda})=\mathbf{1}(\overline{GP}_{\mu})=\mathbf{1}(\overline{GQ}_{\mu})=1$ for all partitions $\lambda$ and all strict partitions $\mu$ .* * Proof* There exists a specialization $\mathbf{1}:\overline{\Gamma}\to\mathbb{R}$ with the formula $\mathbf{1}(\overline{G}_{\lambda})=\lim_{n\to\infty}\overline{G}_{\lambda}(1^{n})=1$ by [24, Rem. 6.10]. This specialization also has $\mathbf{1}(\overline{GP}_{\mu})=\mathbf{1}(\overline{GQ}_{\mu})=1$ by Corollary 3.32. ∎ ### 3.6 Harmonic functions In this final section we describe some applications. Suppose $\mathcal{P}$ is a directed graph with a unique source vertex $\mathbf{0}$ , in which every vertex has finite out-degree, such that there is a finite path from $\mathbf{0}$ to any other vertex $\lambda\in\mathcal{P}$ . We write $\lambda\to\nu$ if there is an edge in $\mathcal{P}$ from a vertex $\lambda\in\mathcal{P}$ to $\nu\in\mathcal{P}$ . A function $\varphi:\mathcal{P}\to\mathbb{R}_{\geq 0}$ is harmonic if $$ \varphi(\mathbf{0})=1\quad\text{and}\quad\varphi(\lambda)=\sum_{\begin{subarray}{c}\nu\in\mathcal{P}\\ \lambda\to\nu\end{subarray}}\varphi(\nu). \tag{𝟎} $$ Let $H(\mathcal{P})$ be the set of harmonic functions $\mathcal{P}\to\mathbb{R}_{\geq 0}$ . This set is convex since if $\varphi_{1},\varphi_{2}\in H(\mathcal{P})$ and $c\in[0,1]$ then $c\varphi_{1}+(1-c)\varphi_{2}\in H(\mathcal{P})$ . Let $\partial H(\mathcal{P})$ be the set extreme points $\varphi\in H(\mathcal{P})$ that cannot be expressed as a convex linear combination of harmonic functions with all nonzero coefficients. **Lemma 3.34** *Suppose there exists an $\mathbb{R}$ -algebra $A$ with basis $\{a_{\lambda}:\lambda\in\mathcal{P}\}$ such that 1. it holds that $a_{\mathbf{0}}=1$ and $a_{\lambda}a_{\mu}\in A_{+}:=\mathbb{R}_{\geq 0}\textnormal{-span}\{a_{\nu}:\nu\in\mathcal{P}\}$ for all $\lambda,\mu\in\mathcal{P}$ , and 1. there exists an index $\mathbf{1}\in\mathcal{P}$ such that $\displaystyle a_{\mathbf{1}}a_{\lambda}=\sum_{\begin{subarray}{c}\nu\in\mathcal{P}\\ \lambda\to\nu\end{subarray}}a_{\nu}$ for each $\lambda\in\mathcal{P}$ . Fix a map $\varphi:\mathcal{P}\to\mathbb{R}$ and let $\rho_{\varphi}:A\to\mathbb{R}$ be the linear map with $\rho_{\varphi}(a_{\lambda})=\varphi(\lambda)$ . Then: 1. $\varphi\in H(\mathcal{P})$ if and only if $\rho_{\varphi}(a_{\lambda})\geq 0$ , $\rho_{\varphi}(a_{\mathbf{0}})=1$ , and $\rho_{\varphi}(a_{\mathbf{1}}a_{\lambda})=\rho_{\varphi}(a_{\lambda})$ for all $\lambda\in\mathcal{P}$ . 1. $\varphi\in\partial H(\mathcal{P})$ if and only if $\rho_{\varphi}$ is a specialization with $\rho_{\varphi}(a_{\lambda})\geq 0$ for all $\lambda\in\mathcal{P}$ and $\rho_{\varphi}(a_{\mathbf{1}})=1$ .* Notice that condition (P2) with $\lambda=\mathbf{0}$ implies that $\mathbf{0}\to\mathbf{1}$ is the unique edge with source $\mathbf{0}$ . * Proof* This lemma may be viewed as a special case of [1, Ex. 4.2]. Alternatively, write $\lambda\vdash n$ if the shortest path from $\mathbf{0}$ to $\lambda\in\mathcal{P}$ has $n$ edges. Then condition (P2) implies that we have $(a_{\mathbf{1}})^{n}-a_{\lambda}\in A_{+}$ and one may deduce the lemma by exactly repeating the proof of [24, Thm. 5.2], replacing the symbols $\tilde{\mathbb{Y}}$ , $\tilde{G}_{(1)}$ , $\tilde{G}_{\lambda}$ , and $\Gamma_{+}$ used there by $\mathcal{P}$ , $a_{\mathbf{1}}$ , $a_{\lambda}$ , and $A_{+}$ . ∎ One may use the preceding lemma to recover several results classifying the extreme harmonic functions on directed graphs associated to integer partitions. Here are two classical examples. Let $\mathbb{Y}$ denote Young’s lattice, the directed graph whose vertices consist of all integer partitions $\lambda$ and whose edges have the form $\lambda\to\nu$ whenever $\lambda\subseteq\nu$ and $|\nu|-|\lambda|=1$ . A specialization $\rho$ of $\mathsf{Sym}$ is normalized if $\rho(h_{1})=1$ (recall that $s_{(1)}=h_{1}$ ). **Corollary 3.35 ([10])** *A map $\varphi:\mathbb{Y}\to\mathbb{R}$ belongs to $\partial H(\mathbb{Y})$ if and only if the linear map $\mathsf{Sym}\to\mathbb{R}$ sending $s_{\lambda}\mapsto\varphi(\lambda)$ is a normalized Schur positive specialization.* * Proof* Apply Lemma 3.34 (using (3.13) with $\beta=0$ ) when $\mathcal{P}=\mathbb{Y}$ , $A=\mathsf{Sym}$ , and $a_{\lambda}=s_{\lambda}$ . ∎ Let $\mathbb{S}\mathbb{Y}$ denote the shifted variant of Young’s lattice, given by the directed graph whose vertices consist of all strict partitions $\lambda$ and whose edges have the form $\lambda\to\nu$ for all strict partitions $\lambda\subseteq\nu$ with $|\nu|-|\lambda|=1$ . A specialization $\varphi$ of $\mathsf{SSym}$ is normalized if $\varphi(h_{1})=1$ (recall that $P_{(1)}=h_{1}$ ). **Corollary 3.36** *A map $\varphi:\mathbb{S}\mathbb{Y}\to\mathbb{R}_{\geq 0}$ belongs to $\partial H(\mathbb{S}\mathbb{Y})$ if and only if the linear map $\mathsf{SSym}\to\mathbb{R}$ sending $P_{\lambda}\mapsto\varphi(\lambda)$ is a normalized Schur $P$ -positive specialization.* * Proof* Apply Lemma 3.34 (using (3.14) with $\beta=0$ ) when $\mathcal{P}=\mathbb{S}\mathbb{Y}$ , $A=\mathsf{SSym}$ , and $a_{\lambda}=P_{\lambda}$ . ∎ Yeliussizov [24, §5] considered the following filtered Young graph $\widetilde{\mathbb{Y}}$ , whose vertices are the same as $\mathbb{Y}$ but which contains an edge $\lambda\to\nu$ whenever $\lambda\subseteq\nu$ and $\mathsf{D}_{\nu}\setminus\mathsf{D}_{\lambda}$ is a nonempty rook strip (that is, having at most one position in each row and column). This is part of the Möbius deformation of $\mathbb{Y}$ in the terminology of [21]. A specialization $\varphi$ of $\Gamma$ is normalized if $\varphi(G_{1})=1$ . The extreme harmonic functions on $\widetilde{\mathbb{Y}}$ are given as follows. **Corollary 3.37 ([24])** *A map $\varphi:\widetilde{\mathbb{Y}}\to\mathbb{R}_{\geq 0}$ belongs to $\partial H(\widetilde{\mathbb{Y}})$ if and only if the linear map $\Gamma\to\mathbb{R}$ sending $G_{\lambda}\mapsto\varphi(\lambda)$ is a normalized $G$ -positive specialization.* * Proof* In view of (3.13) this follows by applying Lemma 3.34 with $\mathcal{P}=\widetilde{\mathbb{Y}}$ , $A=\Gamma$ , and $a_{\lambda}=G_{\lambda}$ . ∎ We introduce a shifted variant: let $\widetilde{\mathbb{S}}\mathbb{Y}$ be the graph with the same vertices as $\mathbb{S}\mathbb{Y}$ but with edges $\lambda\to\nu$ whenever $\lambda\subseteq\nu$ are strict partitions such that $\mathsf{SD}_{\nu}\setminus\mathsf{SD}_{\lambda}$ is a nonempty rook strip. A specialization $\varphi$ of $\Gamma_{P}$ is normalized if $\varphi(GP_{1})=1$ . **Corollary 3.38** *A map $\varphi:\widetilde{\mathbb{S}}\mathbb{Y}\to\mathbb{R}_{\geq 0}$ belongs to $\partial H(\widetilde{\mathbb{S}}\mathbb{Y})$ if and only if the linear map $\Gamma_{P}\to\mathbb{R}$ sending $GP_{\lambda}\mapsto\varphi(\lambda)$ is a normalized $GP$ -positive specialization.* * Proof* In view of (3.14) this follows from Lemma 3.34 with $\mathcal{P}=\widetilde{\mathbb{S}}\mathbb{Y}$ , $A=\Gamma_{P}$ , and $a_{\lambda}=GP_{\lambda}$ . ∎ For convenience, we mention this corollary of Theorem 3.17. 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