Enumeration of H-strata in quantum matrices with respect to dimension

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:119

Article

Bell, J.¹  Casteels, K.²  Launois, S.³ 

[1] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada

[2] Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA

[3] Univ Kent, Sch Math Stat & Actuarial Sci, Canterbury CT2 7NF, Kent, England

关键词: Combinatorics;    Representation theory;    Quantum groups;   

DOI  :  10.1016/j.jcta.2011.07.007

来源: Elsevier

【 摘 要 】

We present a combinatorial method to determine the dimension of H-strata in the algebra of m x n quantum matrices O(q)(M(m,n)(K)) as follows. To a given H-stratum we associate a certain permutation via the notion of pipe dreams. We show that the dimension of the H-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the d-dimensional H-strata in Q(q)(M(m,n)(K)). Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors (Bell and Launois (2010) [3], Bell, Launois and Lutley (2010) [4]) regarding the asymptotic proportion of d-dimensional H-strata in Q(q)(M(m,n) (K)). (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free