Upper generalized exponents of two-colored primitive extremal ministrong digraphs

Generalized exponents of primitive two-colored digraphs

Linear Algebra and its Applications ◽

10.1016/j.laa.2008.03.011 ◽

2009 ◽

Vol 430 (5-6) ◽

pp. 1550-1565 ◽

Cited By ~ 5

Author(s):

Yubin Gao ◽

Yanling Shao

Keyword(s):

Generalized Exponents

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Generalized exponents via Hall-Littlewood symmetric functions

Bulletin of the American Mathematical Society ◽

10.1090/s0273-0979-1987-15519-4 ◽

1987 ◽

Vol 16 (2) ◽

pp. 287-292 ◽

Cited By ~ 17

Author(s):

R. K. Gupta

Keyword(s):

Symmetric Functions ◽

Generalized Exponents

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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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