Absolute order in general linear groups
作者： Victor Reiner , Joel Brewster Lewis , Jia Huang
DOI: 10.1112/JLMS.12013
关键词: Codimension 、 Order (group theory) 、 Character (mathematics) 、 Partially ordered set 、 Mathematics 、 General linear group 、 Additive function 、 Finite field 、 Combinatorics 、 Group (mathematics)
摘要: This paper studies a partial order on the general linear group GL(V) called absolute order, derived from viewing as generated by reflections, that is, elements whose fixed space has codimension one. The is shown to have two equivalent descriptions, one via additivity of length for factorizations into other codimensions. Other properties are derived, including self-duality its intervals. Working over finite field F_q, it complex character computation poset interval identity Singer cycle (or any regular elliptic element) in GL_n(F_q) strikingly simple formula number chains passing through prescribed set ranks.
arxiv.org
128.84.21.199参考文章(29)
Andrey V. Zelevinsky, Representations of Finite Classical Groups Lecture Notes in Mathematics. ,(1981) , 10.1007/BFB0090287
Richard P. Stanley, Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets Electronic Journal of Combinatorics. ,vol. 3, pp. 6- ,(1995) , 10.37236/1264
Joel Brewster Lewis, Alejandro H. Morales, GL n ( F q ) -analogues of factorization problems in the symmetric group The Journal of Combinatorics. ,vol. 58, pp. 75- 95 ,(2016) , 10.1016/J.EJC.2016.05.004
R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field Transactions of the American Mathematical Society. ,vol. 71, pp. 274- 282 ,(1951) , 10.1090/S0002-9947-1951-0043784-0
Victor Reiner, Non-crossing partitions for classical reflection groups Discrete Mathematics. ,vol. 177, pp. 195- 222 ,(1997) , 10.1016/S0012-365X(96)00365-2
Jason Fulman, A Probabilistic Approach Toward Conjugacy Classes in the Finite General Linear and Unitary Groups Journal of Algebra. ,vol. 212, pp. 557- 590 ,(1999) , 10.1006/JABR.1998.7659
Myrto Kallipoliti, The absolute order on the hyperoctahedral group Journal of Algebraic Combinatorics. ,vol. 34, pp. 183- 211 ,(2011) , 10.1007/S10801-010-0267-Z
I.P. Goulden, D.M. Jackson, The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group The Journal of Combinatorics. ,vol. 13, pp. 357- 365 ,(1992) , 10.1016/S0195-6698(05)80015-0
V. Reiner, D. Stanton, D. White, The cyclic sieving phenomenon Journal of Combinatorial Theory, Series A. ,vol. 108, pp. 17- 50 ,(2004) , 10.1016/J.JCTA.2004.04.009
Philippe Biane, Minimal Factorizations of a Cycle and Central Multiplicative Functions on the Infinite Symmetric Group Journal of Combinatorial Theory, Series A. ,vol. 76, pp. 197- 212 ,(1996) , 10.1006/JCTA.1996.0101