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The Green polynomials via vertex operators |
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| Institution: | 1. School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China;2. Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA;1. School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China;2. School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China;3. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China;1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany;2. University of Groningen, the Netherlands, and IMPA, Rio de Janeiro, Brazil;3. Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA |
| Abstract: | An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup. |
| Keywords: | Green's polynomials Vertex operators Hall-Littlewood functions Hecke algebra |
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