[Paper Review] Cluster Variables on Double Bruhat Cells $G^{u,e}$ of Classical Groups and Monomial Realizations of Demazure Crystals

Yuki Kanakubo, Toshiki Nakashima|arXiv (Cornell University)|Apr 20, 2016

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper establishes a monomial realization of Demazure crystals for the non-trivial last $ r $ initial cluster variables on reduced double Bruhat cells $ G^{u,e} $ of classical groups of type $ B_r $, $ C_r $, and $ D_r $. It shows that these generalized minors $ \Delta^L(k;\mathbf{i}) $ are expressed as sums of monomials in specific Demazure crystals, with coefficients greater than one appearing in types $ B_r $, $ C_r $, and $ D_r $, unlike in type $ A_r $. The key contribution is a structural characterization linking cluster algebra variables to crystal lattice realizations beyond type $ A_r $.

ABSTRACT

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ its two opposite Borel subgroups, and $W$ the associated Weyl group. It is shown that the coordinate ring ${\mathbb C}[G^{u,v}]$ ($u$, $v\in W$) of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to the cluster algebra ${\mathcal{A}}( extbf{i})_{\mathbb C}$ and the initial cluster variables of ${\mathbb C}[G^{u,v}]$ are the generalized minors $Δ(k; extbf{i})$ by Berenstein, Fomin, Zelevinsky, Goodearl and Yakimov. In the case that a classical group $G$ is of type ${ m B}_r$, ${ m C}_r$ or ${ m D}_r$, we shall describe the non-trivial last $r$ initial cluster variables $\{Δ(k; extbf{i})\}_{(m-2)r

Motivation & Objective

• To extend the monomial realization of initial cluster variables from type $ A_r $ to classical types $ B_r $, $ C_r $, and $ D_r $.
• To characterize the non-trivial last $ r $ initial cluster variables $ \Delta^L(k;\mathbf{i}) $ in terms of Demazure crystals for reduced double Bruhat cells $ L^{u,e} $.
• To explain the appearance of coefficients greater than one in the monomial expansions, which distinguishes types $ B_r $, $ C_r $, and $ D_r $ from type $ A_r $.
• To provide a structural framework for understanding generalized minors via crystal lattice models, particularly for $ u $ corresponding to left factors of the longest word $ \mathbf{i}_0 $.
• To identify the limitations of Demazure crystal models for certain cluster variables, as shown by counterexamples.

Proposed method

• Uses the reduced double Bruhat cell $ L^{u,e} = (NuN) \cap (B_-eB_-) $ as the geometric setting for cluster algebra structure.
• Applies the isomorphism between the coordinate ring $ \mathbb{C}[L^{u,e}] $ and the cluster algebra $ \mathcal{A}(\mathbf{i})_{\mathbb{C}} $, with initial cluster variables realized as generalized minors $ \Delta(k;\mathbf{i}) $.
• Employs monomial realizations of Demazure crystals via crystal bases of fundamental representations, particularly $ B(\Lambda_r) $, to express $ \Delta^L(k;\mathbf{i}) $ as a sum of monomials.
• Applies Kashiwara's crystal operators $ \tilde{e}_i $ and $ \tilde{f}_i $ to analyze the structure of the crystal lattice and identify the image of the Demazure component under the monomial map $ \mu $.
• Uses recursive application of lowering operators $ \tilde{e}_i $ to reconstruct monomials in $ \mathbb{B}^{(+)}_{sp} $, the set of monomials corresponding to $ \Delta^L(k;\mathbf{i}) $.
• Proves equality $ \mu(B^{-}(\Lambda_r)_{u_{\leq k}}) = \mathbb{B}^{(+)}_{sp} $ via inclusion arguments and explicit operator sequences, establishing the monomial realization.

Experimental results

Research questions

• RQ1 How can the non-trivial last $ r $ initial cluster variables on $ L^{u,e} $ in types $ B_r $, $ C_r $, and $ D_r $ be expressed using crystal lattice models?
• RQ2 Why do coefficients greater than one appear in the monomial expansions of $ \Delta^L(k;\mathbf{i}) $ in types $ B_r $, $ C_r $, and $ D_r $, unlike in type $ A_r $?
• RQ3 To what extent can generalized minors $ \Delta(k;\mathbf{i}) $ be realized as sums over Demazure crystals, and when does this fail?
• RQ4 What is the precise combinatorial structure of the monomials in $ \Delta^L(k;\mathbf{i}) $, and how is it related to the Weyl group element $ u $ and the reduced word $ \mathbf{i} $?
• RQ5 Can a new combinatorial class of crystals be abstracted to match generalized minors beyond Demazure crystals, especially when coefficients exceed one?

Key findings

• The generalized minor $ \Delta^L(k;\mathbf{i}) $ for $ (m-2)r < k \leq (m-1)r $ is expressed as a sum of monomials in a Demazure crystal $ B(\Lambda_r)_{u_{\leq k}} $, with the monomial map $ \mu $ realizing the crystal lattice as a subset of monomials.
• In type $ C_2 $, $ \Delta^L(2;\mathbf{i}) = \frac{Y_{1,1}^2}{Y_{1,2}} + 2\frac{Y_{1,1}}{Y_{2,1}} + \frac{Y_{1,2}}{Y_{2,1}^2} + \frac{1}{Y_{2,2}} $, which matches the sum of all vertices in the crystal graph $ B(\Lambda_2) $ except the highest weight $ Y_{0,2} $, confirming the monomial realization.
• Coefficients greater than one, such as the 2 in $ 2\frac{Y_{1,1}}{Y_{2,1}} $, appear in the expansion due to multiplicity in the crystal lattice, a feature absent in type $ A_r $.
• For $ G $ of type $ C_3 $, $ \Delta^L(3;\mathbf{i}) $ contains a term with coefficient 2 that does not match any Demazure crystal, showing that not all cluster variables are sums over Demazure crystals.
• The set $ \mathbb{B}^{(+)}_{sp} $ of monomials in $ \Delta^L(k;\mathbf{i}) $ is shown to equal $ \mu(B^{-}(\Lambda_r)_{u_{\leq k}}) $, proving the monomial realization via crystal operators.
• The proof relies on constructing a sequence of lowering operators $ \tilde{e}_i $ that map the lowest weight vector $ D(m,r+1) $ to any monomial in $ \mathbb{B}^{(+)}_{sp} $, establishing surjectivity.