Affine type \(A\) geometric crystal on the Grassmannian (Q2318492)

Crystals are combinatorial framework which formalize representations of a Kac-Moody Lie algebra, or a quantum group. They were first introduced by Kashiwara and Luzstig in the study of a canonical basis of integrable modules over a quantum group, called crystal basis. Crystal bases is a canonical bases in which the generators of a quantum group or a Kac-Moody Lie algebra act. They can be regarded as a combinatorial skeleton of a representation of a Kac-Moody Lie algebra. The paper concerns the connection of the geometric crystals of type \(A_{n-1}^{(1)}\) on Grassmannians \(Gr(k,n)\) and the combinatorial one as proposed above. The relation between the geometric crystal and the combinatorics is through the tropical geometry and tropicalization functor. The authors consider certain parametrization of the Grassmannians \(Gr(k,n)\) which can be pulled back through rational functions, where the resulting maps can be tropicalized. The advantage is that they can express certain formulas in terms of Plücker coordinates for crystal operators, [\textit{G. Frieden}, ``The geometric R-matrix for affine crystals of type A'', Preprint, \url{arXiv:1710.07243}; \textit{G. Frieden}, in: Proceedings of the 28th international conference on formal power series and algebraic combinatorics, FPSAC 2016, Vancouver, Canada, July 4--8, 2016. Nancy: The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS). 503--514 (2020; Zbl 1440.05216); \textit{S.-J. Kang} et al., Compos. Math. 92, No. 3, 299--325 (1994; Zbl 0808.17007); \textit{S.-J. Kang} et al., Int. J. Mod. Phys. A 7, 449--484 (1992; Zbl 0925.17005); \textit{M. Kashiwara}, Commun. Math. Phys. 133, No. 2, 249--260 (1990; Zbl 0724.17009); \textit{M. Kashiwara}, Duke Math. J. 63, No. 2, 465--516 (1991; Zbl 0739.17005); \textit{M. Kashiwara} et al., Trans. Am. Math. Soc. 360, No. 7, 3645--3686 (2008; Zbl 1219.17010); \textit{M. Kashiwara} et al., Represent. Theory 14, 446--509 (2010; Zbl 1220.17006); \textit{G. Lusztig}, Prog. Math. 123, 531--568 (1994; Zbl 0845.20034)]. A (combinatorial) crystal is given by a set \(B\) and a collection of maps \begin{itemize} \item \(\widetilde{\gamma}: B \to (\mathbb{Z}_{\geq 0})^n\) \item \(\widetilde{\epsilon}_i: B \to \mathbb{Z}_{\geq 0}, \ i \in \mathbb{Z}/n\). \item \(\widetilde{e}_i, \widetilde{f}_i: B \to (\mathbb{Z}_{\geq 0})^n, \ i \in \mathbb{Z}/n\). \end{itemize} called crystal operators which satisfy certain list of conditions (1--5) [Section 2.1], \begin{itemize} \item[(1)] \(\widetilde{e}_i(b)\) is defined if and only if \(\widetilde{\epsilon}_i(b) > 0\), and when \(e_i(b)\) is defined, \(\widetilde{\epsilon}_i(\widetilde{e}_i(b)) = \widetilde{\epsilon}_i(b)-1\); \item[(2)] \(\widetilde{f}_i(b)\) is defined if and only if \(\widetilde{\phi}_i(b) > 0\), and when \(\widetilde{f}_i(b)\) is defined, \(\widetilde{\phi}_i(\widetilde{f}_i(b))=\widetilde{\phi}_i(b)-1\); \item[(3)] \(\widetilde{e}_i\) and \(\widetilde{f}_i\) are partial inverses, i.e., if \(\widetilde{e}_i(b)\) is defined, then \(\widetilde{f}_i(\widetilde{e}_i(b)) = b\), and if \(\widetilde{f}_i(b)\) is defined, then \(\widetilde{e}_i(\widetilde{f}_i(b)) = b\); \item[(4)] \(\widetilde{\phi}_i(b)- \widetilde{\epsilon}_i(b) = \widetilde{\alpha}_i(\widetilde{\gamma}(b))\), where \(\widetilde{\alpha}_i(a_1, \dots , a_n) = a_i- a_{i+1}\); \item[(5)] If \(\widetilde{e}_i(b)\) is defined, then \(\widetilde{\gamma}(\widetilde{e}_i(b)) = \widetilde{\gamma}(b)+\widetilde{\alpha}_i^{\vee}(1)\), and if \(\widetilde{f}_i(b)\) is defined, then \(\widetilde{\gamma}(\widetilde{f}_i(b)) =\widetilde{\gamma}(b)+\widetilde{\alpha}_i^{\vee}(-1)\), where \(\widetilde{\alpha}_i^{\vee}(m) = m(E_i-E_{i+1})\), with \(E_i\) the \(i\)-th standard basis vector. \end{itemize} A type \(A_{n-1}\) crystal consists of the same data as a type \(A_{n-1}^{(1)}\) crystal, but without the maps associated to \(i = 0\). These operators can be defined on or related to Young Tableaux and partitions, [\textit{D. Bump} and \textit{A. Schilling}, Crystal bases. Representations and combinatorics. Hackensack, NJ: World Scientific (2017; Zbl 1440.17001); \textit{W. Fulton}, Young tableaux. With applications to representation theory and geometry. Cambridge: Cambridge University Press (1997; Zbl 0878.14034); \textit{B. Rhoades}, J. Comb. Theory, Ser. A 117, No. 1, 38--76 (2010; Zbl 1230.05289); \textit{M. P. Schützenberger}, Discrete Math. 2, 73--94 (1972; Zbl 0279.06001)]. It leads also to define \begin{itemize} \item the promotion \(\widetilde{pr}=\widetilde{\sigma}_1 \dots \widetilde{\sigma}_{n-1}\) and \item the Schutzenberger involution \(S=(\widetilde{\sigma}_1) (\widetilde{\sigma}_2 \widetilde{\sigma}_1)\dots (\widetilde{\sigma}_{n-1} \dots \widetilde{\sigma}_1)\) \end{itemize} where \(\widetilde{\sigma}_i\) is the involution that exchanges the \(i\)-th and \((i-1)\)-th number in a semistandard Young Tableaux (SSYT) [see Section 2.2.1]. We consider the following notations in the text \begin{itemize} \item A Gelfand-Tsetlin pattern is a triangle \((A_{ij})_{1 \leq i \leq j \leq n}\) such that \(A_{i,j+1} \geq A_{ij} \geq A_{i+1,j+1}\). One associates a Tableaux \(T\) to a Gelfand-Tsetlin pattern so that the number of \(j\) in the \(i\)-th row of \(T\) is \(A_{ij}-A_{i,j-1}\). \item \(R_k=\{(i,j)|1 \leq i \leq k, i \leq j \leq n-k \}, \widetilde{\mathbb{T}}_k=\mathbb{Z}^{R_k} \times \mathbb{Z}\) where denote the elements in \(\widetilde{\mathbb{T}}_k\) by \((B_{ij},L)\). Given \((B_{ij},L)\), lets define \[ A_{ij}=\begin{cases}B_{ij}, \qquad \ \ \ \ (i,j) \in R_k\\ A_{ij}=L \qquad j \geq i+n-k\\ A_{ij}=0 \qquad j<i \end{cases} \] Set \(B^k\) to be the \((B_{ij},L)\) such that \((A_{ij})\) defines a Gelfand-Tsetlin pattern. One has \[ B^k= \bigsqcup_k B^{k,L} \] \item Define \(\widetilde{rot}:B^k \to B^k, \widetilde{rot}(B_{ij},L)=(C_{ij},L)\) where \(C_{ij}=L-B_{k-i+1,n-j}\), and \(\widetilde{refl}:B^k \to B^k, \widetilde{refl}(B_{ij},L)=(D_{ij},L)\) where \(D_{ij}=L-B_{j-i+1,j}\). \end{itemize} The relations between these maps are given in Proposition 2.15, \begin{align*} \widetilde{e}_i=\widetilde{rot} \circ \widetilde{f}_{n-i} \circ \widetilde{rot}\\ \hat{e}_i=\widetilde{refl} \circ \widetilde{f}_{i} \circ \widetilde{refl} \end{align*} A geometric pre-crystal of type \(A_{n-1}^{(1)}\) is a tuple \((X, \gamma, \phi_i, \epsilon_i, e_i)\) where \(X\) is an algebraic variety, \(\gamma:X \to (\mathbb{C}^{\times})^n\). is a rational map, \(\phi_i, \epsilon_i:X \to \mathbb{C}^{\times}\) are not-identically zero maps, and \(e_i:\mathbb{C}^{\times} \times X \to X\) is a rational unital action, \(i \in \mathbb{Z}/n\). These maps have to satisfy the following conditions \begin{itemize} \item \(\epsilon_i(x)=\phi_i(x)\alpha_i(\gamma(x))\) where \(\alpha_i(z_1,\dots ,z_n)=z_i/z_{i+1}\). \item \(\gamma(e_i^c(x))=\alpha_i^{\vee}(c)\gamma(x)\) where \(\alpha_i^{\vee}(1,\dots ,c,c^{-1},\dots ,1)\) where \(c\) is in the \(i\)-th component \(\mod n\). \item \(\phi_i(e_i^c(x))=c^{-1}\phi_i(x), \epsilon_i(e_i^c(x))=c\epsilon_i(x)\). \end{itemize} A geometric crystal is a precrystal where the the operations \(e_i^c\) satisfy the Serre relations [definition 3.5]. That is If \(n \geq 3\), then for each pair of distinct \(i, j \in \mathbb{Z}/n\), the actions \(e_i, e_j\) satisfy \begin{align*} e_i^{c_1}e_j^{c_2}&= e_j^{c_2} e_i^{c_1} \qquad \qquad i- j\equiv \pm 1 \mod n\\ e_i^{c_1}e_j^{c_1c_2}e_i^{c_2}&= e_j^{c_2} e_i^{c_1c_2} e_j^{c_1} \qquad i - j \equiv \pm 1 \mod n \end{align*} for all \(c_1, c_2 \in \mathbb{C}^{\times}\) such that both sides are defined. Moreover a geometric crystal maybe equipped with a decoration function \(f:X \to \mathbb{C}^{\times}\) satisfying the following equation along the action of \(e_i^c\) [Definition 3.6] \[ f(e_i^c(x)) = f(x) + \frac{c-1}{\phi_i(x)} + \frac{c^{-1}-1}{\epsilon_i(x)} \] In case of \(X =\mathbb{X}_k=Gr(k,n) \times \mathbb{C}^{\times}\) the maps \(\gamma, \epsilon_i, \phi_i, f\) may be given in terms of the Plucker coordinates. We represent a point \(M \in Gr(k, n)\) as the column span of a (full-rank) \(n \times k\) matrix \(M'\), so that \(P_J (M)\) is the maximal minor of \(M'\) using the rows in \(J\). We list the definition of the crystal maps as follows, [see Definition 3.7] \begin{itemize} \item \begin{align*} \gamma&: \mathbb{X}_k \to (\mathbb{C}^{\times})^n, \ \ \gamma(M,t) = (\gamma_1, \dots , \gamma_n), \\ \gamma_i &=\begin{cases}\dfrac{P_{[i-k+1,i]}(M)}{P_{[i-k,i-1]}(M)}, \qquad \ \ 1 \leq i \leq k\\ t. \dfrac{ P_{[i-k+1,i]}(M)}{P_{[i-k,i-1]}(M)} , \qquad k + 1 \leq i \leq n \end{cases} \end{align*} \item \begin{align*} \phi_i, \epsilon_i \ \ &: \mathbb{X}_k \longrightarrow \mathbb{C}^{\times},\\ \phi_i(M,t) &= t^{\delta_{i,0}} \frac{P_{[i-k+1,i-1]\cup \{i+1\}}(M)}{P_{[i-k+1,i]}(M)} ,\\ \epsilon_i(M,t) &= t^{-\delta_{i,k}} \frac{P_{[i-k+1,i-1]\cup \{i+1\}}(M)P_{[i-k+1,i]}(M)}{P_{[i-k,i-1]}(M)P_{[i-k+2,i+1]}(M)} \end{align*} \item \begin{align*} e_i^c&: \mathbb{X}_k \to \mathbb{X}_k, \ \ e_i^c(M,t) = (M',t), \\ M' &=\begin{cases}x_i \left (\dfrac{c-1}{\phi_i(M,t)}\right ) M, \qquad \qquad \qquad i \ne 0\\ x_0 \left (\dfrac{(-1)^{k-1}}{t}. \dfrac{c-1}{\phi_0(M,t)}\right )M \ \qquad i = 0 \end{cases}\\ x_i(a) &= I + aE_{i,i+1}, \qquad i \in [n-1], \\ x_0(a) &= I + aE_{n1} \end{align*} \item \[ f(M,t) = \sum_{i \ne k}\frac{P_{\{i-k\}\cup [i-k+2,i]}(M)}{P_{[i-k+1,i]}(M)} + t. \frac{P_{[2,k] \cup \{n\}}(M)}{P_{[1,k]}(M)} \] \end{itemize} The main theorem of Section 3 states that \((\mathbb{X}_k, \gamma, \phi_i, \epsilon_i, e_i, f)\) form a decorated geometric crystal. In this case the set of maps \(\phi_i\) and also \(\epsilon_i\) can be related through the shifting operation \(PR:\mathbb{X}_k \to \mathbb{X}_k\) defined by \(PR(M,t) = (M',t)\), where \(M'\) is obtained from \(M\) by shifting the rows down by \(1 (mod \ n)\), and multiplying the new first row by \((-1)^{k-1}t\). The result comes in Lemma 3.11, \begin{itemize} \item \(\phi_i \circ PR=\phi_{i-1}\) and, \(\epsilon_i \circ PR=\epsilon_{i-1}\). \item \(PR^{-1} \circ e_i^c \circ PR=e_{i-1}^c\). \item If \(\gamma=(\gamma_1,\dots ,\gamma_n)\) then \(\gamma \circ PR=(\gamma_n,\gamma_1,\dots ,\gamma_{n-1})\). \item \(f \circ PR=f\). \end{itemize} In Section 4 the authors consider a parametrization of \(\mathbb{X}_k\). Denote \(\mathbb{T}_k=(\mathbb{C}^{\times})^{R_k} \times \mathbb{C}^{\times}\) where the points we denote by \((X_{ij},t)\). Set also \(x_{ij}=X_{ij}/X_{i,j-1}\). Then define \begin{align*} \Phi_{n-k} \ &:\mathbb{T}_{n-k} \longrightarrow GL_n\\ \Phi_{n-k}(X_{ij},t)&=\prod_{i=n-k}^l M_{[i,i+k]}(x_{ii},x_{i,i+1},\dots ,x_{i,i+k}) \end{align*} where \[ M_{[a,b]}(z_a,\dots ,z_b) = \sum_{i \in [a,b]} z_iE_{ii} + \sum_{i \in [n]\setminus [a,b]} E_{ii} + \sum_{i \in [a+1,b]}E_{i,i-1}, \] for \(1 \leq a,b \leq n, z_j \in \mathbb{C}^{\times}\). They define the parametrizations \(\Theta_{n-k}:\mathbb{T}_{n-k} \to \mathbb{X}_k\) by sending \((X_{ij},t)\) to \((M,t)\) where \(M\) is the subspace spanned by the first \(k\)-collumns of \(\Phi_{n-k}(X_{ij},t)\). Proposition 4.3 calculates the inverse of \(\Theta_k\) in terms of Plucker coordinates; \((M,t) \mapsto (X_{ij} ,t)\), where \[ X_{ij} = \dfrac{P_{[i,j] \cup [n-k+j-i+2,n]}(M)}{P_{[i+1,j]\cup [n-k+j-i+1,n]}(M)}, \qquad 1 \leq i \leq n-k,\ i \leq j \leq i + k-1 \] A fundamental definition in Section 4 is that of basic Plucker coordinates \(P_{J_{i,j}}\) where \(J_{i,j}=[i,j] \cup [n-k+j-i+2,n]\) given in Definition 4.17. Lemma 4.18 states that elements in an open \(U_k \subset Gr(k,n)\) such that all its coordinates are non-zero are uniquely determined by their basic Plucker coordinates. By Corollary 4.19 Every Plucker coordinate can be written as a Laurent polynomial in the basic Plucker coordinates with non-negative integer coefficients. As mentioned before the relation between geometric and combinatorial crystals is through tropical geometry and tropicalization functor, [\textit{A. Berenstein} et al., Adv. Math. 122, No. 1, 49--149 (1996; Zbl 0966.17011); \textit{A. Berenstein} and \textit{A. Zelevinsky}, Duke Math. J. 82, No. 3, 473--502 (1996; Zbl 0898.17006); Invent. Math. 143, No. 1, 77--128 (2001; Zbl 1061.17006); \textit{A. N. Kirillov}, in: Physics and combinatorics. Proceedings of the Nagoya 2000 2nd international workshop, Nagoya, Japan, August 21--26, 2000. Singapore: World Scientific. 82--150 (2001; Zbl 0989.05127)]. The parametrizations \(\Theta_r\) can be composed with functions on \(\mathbb{X}_k\) or the maps between them. In case the coordinate components of these functions are subtraction free (i.e. their coefficients lie in the tropical semifield) they can be tropicalized. Define \( \widehat{\gamma}=Trop(\gamma^{\Theta}), \widehat{\phi}_i=Trop(\phi_i^{\Theta}), \widehat{\epsilon}_i=Trop(\epsilon_i^{\Theta})\) where we are using the notation at the beginning of Section 5.2.1 [superfix means composition with \(\Theta_{n-k}\)]. Theorems 5.4 and 5.7 are the main results in Section 5 stating that \begin{itemize} \item \(\widehat{\gamma}_i=\widetilde{\gamma}_i\), \item \(\widehat{\phi}_i=-\widetilde{\phi}_i\) and, \(\widehat{\epsilon}_i=-\widetilde{\epsilon}_i\), \item \(\widehat{e}_i(1,b)=\widetilde{e}_i\) and, \( \widehat{\epsilon}_i(-1,b)=\widetilde{f}_i\), where \(\widehat{e}_i=Trop(\Theta_{n-k}^{-1} \circ e_i^c \circ \Theta_{n-k})\) when all the actions are defined. \item \(\widehat{PR}=\widetilde{pr}\) where \(\widehat{PR}=Trop(\Theta_{n-k}^{-1} \circ PR \circ \Theta_{n-k})\). \end{itemize} Section 7 discusses symmetries and dualities of crystal operators. It briefly purpose that the symmetries descend from the geometric case to the combinatorial one by tropicalization. The symmetries are also compatible by the duality on the Grassmannian in a natural way. Assume \(S:\mathbb{X}_k \to \mathbb{X}_k\) is given by [Definition 7.1], \[ S(M,t) =(M',t), \qquad M' = \pi_t^k \circ fl \circ g_t(M). \] where \(\pi_t^k\) and \(fl\) are defined in Section 7.1, and \(g_t\) is given by Definition 6.7 [the definitions are in terms of folding a finite matrix and conversely the unfolding the infinite one, explained in Section 6, see also [\textit{A. Berenstein} and \textit{D. Kazhdan}, in: GAFA 2000. Visions in mathematics---Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part I. Basel: Birkhäuser. 188--236 (2000; Zbl 1044.17006); Contemp. Math. 433, 13--88 (2007; Zbl 1154.14035)]. Let \(Q_{w_0}\) be the permutation matrix corresponding to the longest element of \(S_n\). Define \(T_{w_0} : Gr(k, n) \to Gr(k, n)\) by \(T_{w_0} (M) = Q_{w_0}.M\). Then define the geometric contragredient duality map \(D:\mathbb{X}_k \to \mathbb{X}_{n-k}\) by \(D(M,t) = S(T_{w_0}(M^{\perp}), t)\) [see Definition 7.6 in Section 7.2]. The following list of identities are mentioned in the Corollaries 7.4 and 7.9. \begin{itemize} \item \(S^2=id\) and \(S \circ PR=PR \circ S\). \item \(\phi_i \circ S=\epsilon_{n-i}\) and \(\epsilon_i \circ S=\phi_{n-i}\). \item \(S \circ e_i^c=e_{n-i}^{c^{-1}} \circ S\) \item \(S \circ D=D \circ S\) and \(PR \circ D=D \circ PR\). \item \(\phi_i \circ D=\epsilon_i\) and, \(\epsilon_i \circ D=\phi_i\). \item \(e_i^c \circ D=D \circ e_i^{c^{-1}}\). \end{itemize} One has the tropical analogue of \(S\) and \(D\) defined as \begin{align*} \widehat{S}&=Trop(\Theta_{n-k}^{-1} \circ S \circ \Theta_{n-k}):\widetilde{\mathbb{T}}_k \to \widetilde{\mathbb{T}}_k\\ \widehat{D}&=Trop(\Theta_{k}^{-1} \circ D \circ \Theta_{n-k}):\widetilde{\mathbb{T}}_k \to \widetilde{\mathbb{T}}_{n-k} \end{align*} By Theorem 7.10 \begin{itemize} \item \(\widehat{S}=\widetilde{rot}\), \item \(\widehat{D}=\widetilde{refl}\). \end{itemize}