Combinatorial methods of character enumeration for the unitriangular group

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• 作者：Eric  ; Marberg¹ ;   ; ^{emarberg@math.mit.edu}
• 关键词：Algebra groups ; Unitriangular group ; Irreducible characters ; Supercharacters ; Kirillov functions ; Set partitions ; Connected set partitions ; Lehrer?s conjecture
• 刊名：Journal of Algebra
• 出版年：2011
• 出版时间：1 November 2011
• 年：2011
• 卷：345
• 期：1
• 页码：295-323
• 全文大小：393 K

文摘

Let denote the group of unipotent upper triangular matrices over a field with q elements. The degrees of the complex irreducible characters of are precisely the integers with , and it has been conjectured that the number of irreducible characters of with degree is a polynomial in with nonnegative integer coefficients (depending on n and e). We confirm this conjecture when and n is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in n and q giving the number of irreducible characters of with degree when and . When divided by and written in terms of the variables and , these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of with degree are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of .