Jacob Greenstein
We construct a subcrystal of the Littelmann's path crystal whose formal character coincides with that of a certain simple integrable module of level zero over the untwisted affine Lie algebra associated to sl_n. We also establish an analogue of the Littlewood-Richardson rule for the tensor product of that crystal with a highest weight crystal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We give a new model for the crystal graphs of an affine Lie algebra g^, combining Littelmann's path model with the Kyoto path model. The vertices of the crystal graph are represented by certain infinitely looping paths which we call skeins. We apply this model to the case when the corresponding finite-dimensional algebra g has a minuscule representation (classical type and E_6, E_7). We prove that the basic level-one representation of g^, when considered as a representation of g, is an infinite tensor product of fundamental representations of g. A similar tensor product phenomenon holds for ce…
A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara'…
For a certain class of simple integrable modules of level zero over a quantised affine algebra, we establish the existence of a pseudo-crystal basis and show that such a basis admits a combinatorial realisation in the framework of the path model of Littelmann.
In this paper, we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The characters of basic representations c…
We present an explicit combinatorial realization of the commutor in the category of crystals which was first studied by Henriques and Kamnitzer. Our realization is based on certain local moves defined by van Leeuwen.
Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of $U_q(\widehat{\mathfrak{sl}}_\ell)$-generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Eul…
WKB approximation, crystals and combinatorics of Young tableaux
该研究揭示了量子合流超几何方程连接矩阵的WKB近似可自然衍生出杨表的多种组合算法，包括Robinson-Schensted算法、Littlewood-Richardson规则与Schützenberger对合。
Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers
本文建立了A型和D型单李代数晶体基两种实现间的显式晶体同构，给出了Lusztig箭图簇不可约分支的同调描述，并通过A型提升算子实现了Kirillov-Reshetikhin晶体的几何构造。
Crystal skeletons: Combinatorics and axioms
该研究针对晶体骨架的相关问题，给出其组合描述、证明新性质（含Maas-Gariépy猜想），并提出基于$GL_n$-分支、$S_n$-分支及局部的三类公理体系，推动Schur正性相关研究。
We explore combinatorics associated with the degenerate Hecke algebra at $q=0$, obtaining a formula for a system of orthogonal idempotents, and also exploring various pattern avoidance results. Generalizing constructions for the 0-Hecke algebra, we explore the representation theory of $\JJ$-trivial monoids. We then discuss two-tensors of crystal bases for $U_q(\tilde{\mathfrak{sl}_2})$, establishing a complementary result to one of Bandlow, Schilling, and Thiéry on affine crystals arising from promotion operators. Finally, we give a computer implementation of Stembridge's local axioms for simp…
We show that Joseph-Lamprou's path model for representations of generalized Kac-Moody algebras can be embedded into Littelmann's path model for certain Kac-Moody algebras. Using this embedding, for Joseph-Lamprou's path crystals, we give a decomposition rule for tensor product and a branching rule for restriction to Levi subalgebras. Also, we obtain a characterization of standard paths in terms of a certain monoid, which can be thought of as a generalization of a Coxeter group.
Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(λ)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$. The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the constructions of crystal graphs.
We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…
We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of our previous result (which uses Littelmann's Lakshmibai-Seshadri paths), in which the level-zero dominant integral weight is assumed to be a posi…
Kang et al. provided a path realization of the crystal graph of a highest weight module over a quantum affine algebra, as certain semi-infinite tensor products of a single perfect crystal. In this paper, this result is generalized to give a realization of the tensor product of several highest weight modules. The underlying building blocks of the paths are finite tensor products of several perfect crystals. The motivation for this work is an interpretation of fermionic formulas, which arise from the combinatorics of Bethe Ansatz studies of solvable lattice models, as branching functions of affi…
We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively .
A new Young wall realization of $B(λ)$ and $B(\infty)$
本文利用 Young 墙的新组合学，为量子仿射代数构造了任意水平最高权晶体 $B(λ)$ 和 $B(\infty)$ 的新实现，并证明了约化 Young 墙晶体与 $B(λ)$ 同构。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。