Chapter Eighteen. Statement B and Crystal Graphs

This chapter translates Statements A and B into Statements A′ and B′ in the language of crystal bases, and explains in this language how Statement B′ implies Statement A′. It first introduces the relevant definition, which is provisional since it assumes that we can give an appropriate definition of boxing and circling for Ω‎. The crystal graph formulation in Statement A′ is somewhat simpler than its Gelfand-Tsetlin counterpart. In particular, in the formulation of Statement A, there were two different Gelfand-Tsetlin patterns that were related by the Schützenberger involution. In the crystal graph formulation, different decompositions of the long element simply result in different paths from the same vertex v to the lowest weight vector.

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Quivers with relations for symmetrizable Cartan matrices IV: crystal graphs and semicanonical functions

Selecta Mathematica ◽

10.1007/s00029-018-0412-4 ◽

2018 ◽

Vol 24 (4) ◽

pp. 3283-3348

Author(s):

Christof Geiss ◽

Bernard Leclerc ◽

Jan Schröer

Keyword(s):

Crystal Graphs ◽

Cartan Matrices

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Combinatorics of Generalized Exponents

International Mathematics Research Notices ◽

10.1093/imrn/rny157 ◽

2018 ◽

Vol 2020 (16) ◽

pp. 4942-4992 ◽

Cited By ~ 1

Author(s):

Cédric Lecouvey ◽

Cristian Lenart

Keyword(s):

Root System ◽

Lie Algebras ◽

Finite Type ◽

Irreducible Representations ◽

Combinatorial Proof ◽

Combinatorial Formula ◽

Branching Rules ◽

Crystal Graphs ◽

Type C ◽

Generalized Exponents

Abstract We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type $A_{n-1}$, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type $C_{n}$, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type $A_{2n-1}$, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig $t$-analogs associated to zero-weight spaces in the irreducible representations of symplectic Lie algebras. We also present three applications of our combinatorial formula and discuss some implications to relating two type $C$ branching rules. Our methods are expected to extend to the orthogonal types.

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