The combinatorics of GL n generalized Gelfand-Graev characters

Andrews, Scott; Thiem, Nathaniel

doi:10.1112/jlms.12023

Cited by 8 publications

(7 citation statements)

References 16 publications

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“…This section introduces several facts about the characters of GL n (F q ), and uses them to define and understand the main characters we need in the paper. We follow the terminology of [AT17].…”

Section: The Corresponding Epimorphism Is Denoted By Pmentioning

confidence: 99%

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A Generation Criterion for Subsets of $SL_n(\mathbb{F}_q)$

Greenhut¹

2020

Preprint

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Let G 0 be a either SL n (F q ), the special linear group over the finite field with q elements, or PSL n (F q ), its projective quotient, and let Σ be a symmetric subset of G 0 , namely, if x ∈ Σ then x −1 ∈ Σ. We find a certain set R(G 0 ) of irreducible representations of G 0 whose size is at most 5, such that Σ generates G 0 if and only if |Σ| is not an eigenvalue of σ∈Σ ρ(σ) for every ρ ∈ R(G 0 ).To achieve this result, let G be either GL n (F q ) or PGL n (F q ). We consider X (G), some set of irreducible nontrivial characters of G, whose size is at most 5. We show that for every subgroup K ≤ G that does not contain G 0 , the restriction to K of at least one of the characters in X (G) contains the trivial character as an irreducible summand. We then restrict the characters to G 0 and use standard arguments about the Cayley graph of G to imply the result. In addition, we obtain slightly weaker results about the generation of symmetric subsets of G.We finish by considering S n , the symmetric group on n elements, and presenting R(S n ), a set of eight irreducible nontrivial representations of S n , such that a symmetric subset Σ ⊆ S n generates S n if and only if |Σ| is not an eigenvalue of σ∈Σ ρ(σ) for every ρ ∈ R(S n ), which is an improvement upon the previously known set of 12 irreducible nontrivial representations of S n that satisfies this condition.

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Section: The Corresponding Epimorphism Is Denoted By Pmentioning

confidence: 99%

“…In[AT17], these characters are labeled χ µ ′(1) . We define: χ µ = χ µ ′(1) , which is more similar to the notation of[Gre55].…”

mentioning

confidence: 99%

A Generation Criterion for Subsets of $SL_n(\mathbb{F}_q)$

Greenhut¹

2020

Preprint

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“…In the case of GL n (F q ), the nilpotent orbits are indexed by the partitions of n; let Γ λ be the generalized Gelfand-Graev character indexed by the partition λ. In [1], Thiem and the author show that Γ λ can be obtained by inducing a linear character from U λ ′ , and calculate the multiplicities of the unipotent characters in Γ λ .…”

Section: Proposition 33 ([11]mentioning

confidence: 99%

“…(1 1), where P λ ′ is the parabolic subgroup of GL n (F q ) of shape λ ′ (see Section 1). There are two reasonable analogues to Ind Sn W λ ′ (ǫ), however; these are the "degenerate Gelfand-Graev characters" of Zelevinsky [11] and the "generalized Gelfand-Graev characters," which were initially constructed by Kawanaka [8] and recently studied by Thiem and the author [1]. James' construction in [6] uses the degenerate Gelfand-Graev characters; we instead utilize the generalized Gelfand-Graev characters.…”

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confidence: 99%

The irreducible unipotent modules of the finite general linear groups via tableaux

Andrews

2015

Preprint

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We construct the irreducible unipotent modules of the finite general linear groups using tableaux. Our construction is analogous to that of James (1976) for the symmetric groups, answering an open question as to whether such a construction exists. Our modules are defined over any field containing a nontrivial p th root of unity (where p is the defining characteristic of the group). We show that our modules are isomorphic to those constructed by James (1984), although the two constructions utilize different approaches. Finally we look closer at the complex irreducible unipotent modules, providing motivation for our construction in the language of symmetric functions.

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“…While they were a fundamental example in [13], the supercharacter theory they give in that paper for pattern groups is not generally well understood. Andrews introduced a different supercharacter theory called a non-nesting supercharacter theory that has nice combinatorial properties [8]; in fact, [9] used this theory to study generalize Gelfand-Graev characters for the finite general linear groups. While his theory differs from ours, it is in fact morally equivalent.…”

Section: Introductionmentioning

confidence: 99%

Pattern groups and a poset based Hopf monoid

Aliniaeifard

Thiem

2020

Journal of Combinatorial Theory, Series A

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The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

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The combinatorics of GL n generalized Gelfand-Graev characters

Cited by 8 publications

References 16 publications

A Generation Criterion for Subsets of $SL_n(\mathbb{F}_q)$

A Generation Criterion for Subsets of $SL_n(\mathbb{F}_q)$

The irreducible unipotent modules of the finite general linear groups via tableaux

Pattern groups and a poset based Hopf monoid

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