Counting irreducible representations of large degree of the upper triangular groups

Le, Tung

doi:10.1016/j.jalgebra.2010.06.022

Cited by 10 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is conceivable that similar PORC (Polynomial On Residue Classes) behaviour occurs for k(U (q)) if G is of type E 8 . For other recent developments, see for example [13,16,17,20].…”

Section: Introductionmentioning

confidence: 99%

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

Goodwin¹,

Mosch²,

Röhrle³

2014

LMS J. Comput. Math.

View full text Add to dashboard Cite

Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U (q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U (q)) of conjugacy classes in U (q) is given by a polynomial in q with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321-3334], the first and the third authors of the present paper developed an algorithm to calculate the values of k(U (q)). By implementing it into a computer program using GAP, they were able to calculate k(U (q)) for G of rank at most five, thereby proving that for these cases k(U (q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U (q)) for finite Chevalley groups of rank six and seven, except E7. We observe that k(U (q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U (q)) as a polynomial in q − 1, then the coefficients are non-negative.Under the assumption that k(U (q)) is a polynomial in q − 1, we also give an explicit formula for the coefficients of k(U (q)) of degrees zero, one and two.

show abstract

“…It is conceivable that similar PORC (Polynomial On Residue Classes) behaviour occurs for k(U (q)) if G is of type E 8 . For other recent developments, see for example [13,16,17,20].…”

Section: Introductionmentioning

confidence: 99%

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

Goodwin¹,

Mosch²,

Röhrle³

2014

LMS J. Comput. Math.

View full text Add to dashboard Cite

show abstract

“…Supercharacters arise as tensor products of some elementary characters to give a 'nice' partition of all non-principal irreducible characters of U n (q) (see [1,12]). Supercharacters arise as tensor products of some elementary characters to give a 'nice' partition of all non-principal irreducible characters of U n (q) (see [1,12]).…”

Section: Introductionmentioning

confidence: 99%

“…Many efforts have been made to understand more about U n (q); see [1,3,5,7,10,11,14,15], among others. Supercharacters arise as tensor products of some elementary characters to give a 'nice' partition of all non-principal irreducible characters of U n (q) (see [1,12]). Supercharacters have been defined for Sylow p-subgroups of other finite groups of Lie type (see [2]), and in general for algebra groups (see [5]).…”

Section: Introductionmentioning

confidence: 99%

Supercharacters and pattern subgroups in the upper triangular groups

2012

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Immediately, we have this corollary: Our proof of the theorem will follow from two short lemmas, which we state below in rapid succession. The first of these is an immediate consequence of [25,Lemma 3.4]; we provide a short proof using Lemmas 2.1 and 2.2 for completeness. Proof.…”

Section: Factorizations Of N λ (Q) and N λE (Q)mentioning

confidence: 90%

“…In his paper [21], Isaacs contributes some additional formulas. More recently, Loukaki [27] has computed N n,e (q) when 0 ≤ e ≤ 3, and Le [25] has rederived Marjoram's formulas for N n,e (q) when M n − 2 ≤ e ≤ M n (this part of Marjoram's work was never published and required n to be even; Le removes this condition).…”

mentioning

confidence: 99%

Combinatorial methods of character enumeration for the unitriangular group

Marberg¹

2011

Journal of Algebra

View full text Add to dashboard Cite

Let UT n (q) denote the group of unipotent n × n upper triangular matrices over a field with q elements. The degrees of the complex irreducible characters of UT n (q) are precisely the integers q e with 0 ≤ e ≤ ⌊ n 2 ⌋⌊ n−1 2 ⌋, and it has been conjectured that the number of irreducible characters of UT n (q) with degree q e is a polynomial in q−1 with nonnegative integer coefficients (depending on n and e). We confirm this conjecture when e ≤ 8 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 UT n (q) with degree q e when n > 2e and e ≤ 8. When divided by q n−e−2 and written in terms of the variables n − 2e − 1 and q − 1, 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 UT n (q) with degree ≤ q 8 are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of UT n (q). IntroductionLet F q be a finite field with q elements and write UT n (q) to denote the unitriangular group of n × n upper triangular matrices over F q with all diagonal entries equal to 1. This is a Sylow p-subgroup of the general linear group GL(n, F q ), where p > 0 is the characteristic of F q . This work concerns the problem of counting the irreducible characters of UT n (q).By a result of Isaacs [20], the degrees of all (complex) irreducible characters of UT n (q) are powers of q. In fact, Huppert [19] has shown that the set of integers occurring as degrees of irreducible characters of UT n (q) is {q e : 0 ≤ e ≤ M n }, whereOne may therefore define N n (q) and N n,e (q) for each positive integer n, prime power q > 1, and integer e, as the numbers N n (q) = the number of irreducible characters (also, of conjugacy classes) of UT n (q), N n,e (q) = the number of irreducible characters of UT n (q) of degree q e .

show abstract

Counting irreducible representations of large degree of the upper triangular groups

Cited by 10 publications

References 10 publications

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

Supercharacters and pattern subgroups in the upper triangular groups

Combinatorial methods of character enumeration for the unitriangular group

Contact Info

Product

Resources

About