Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Dipper, Richard; Guo, Qiong

doi:10.1016/j.jpaa.2014.09.016

Cited by 8 publications

(11 citation statements)

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“…unipotent groups). Not only can these new theories be combinatorially striking [1], but they can also be used in place of the usual character theory [6] in applications, they give a starting point in studying difficult theories [12], or give character theoretic foundations for number theoretic identities (eg. [8,14]).…”

Section: Introductionmentioning

confidence: 99%

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

Thiem¹

2018

Algebraic Combinatorics

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Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of the combinatorial properties of the set partition combinatorics of the full uni-triangular groups, including combinatorial indexing sets, dimensions, and computable character formulas. Associated with these supercharacter theories is also a family of polytopes whose integer lattice points give the theories geometric underpinnings. PreliminariesThis paper is primarily concerned with unipotent subgroups of the finite general linear groups. It is standard to index the rows and columns of the corresponding matrices by the set t1, 2, . . . , N u in the usual total order, but recent work [2] has suggested that it is better to instead allow an arbitrary set of size N to index the rows and columns and fix a total order N on that set. However, the paper may be easily read with N as the total order 1 ă 2 ă¨¨¨ă N .This section reviews the relevant unipotent groups, a combinatorial interpretation of normality, and some of the standard supercharacter theories for these groups. The last section gives a quick refresher of q-binomial coefficients.

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Section: Introductionmentioning

confidence: 99%

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

Thiem¹

2018

Algebraic Combinatorics

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“…We observe that in example 5.3 the J-places on the hook arm centered at z 1 are the positions = (1, 5), (1,6), (1,7) and are in bijection by the flip map 4.3 with the non normal positions on the hook leg centered at the main condition (1,8). These are all positions on column 8 south of (1,8) except the normal positions (5,8), (6,8) and (7,8). All of those, except three main hook intersections (2,8), (3,8) and (4,8), are Y-conditions and hence contribute q a to the local κ, where a is the number of these Y-conditions in column 8.…”

Section: A Guiding Examplementioning

confidence: 84%

“…where [Ã] ∈V is the template which agrees with [A] at all positions except 3 supplementary normal conditions at positions (5,8), (6,8), (7,8), which are given as λ, ρ and the third term dependent on λ, ρ. Then CO r A ∼ = CO rÃ and by the free choices of λ and ρ, we have q 2 many different right orbits shown to be isomorphic.…”

Section: A Guiding Examplementioning

confidence: 99%

“…Now the constellation of the normal supplementary conditions to the right of column 8 denoted by Greek letters determines hook intersection (1,2) to contribute a factor q to automorphisms of O r A , that is to O r A ∩ O l A . The normal positions (5,8) and (6,8) in exchange for the remaining two hook intersections (2,8) and (3,8) not yet accounted for, contribute factors q to κ in 5.5. Thus for the J-places in row 1, that is the number of left J-places in column 8 equation 5.5 reads now q |Y(p,J)| 8 · q 2 · q = q |leftJ-places| 8…”

Section: A Guiding Examplementioning

confidence: 99%

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On monomial linearisation and supercharacters of pattern subgroups

Guo

Dipper

2017

Sci. China Math.

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Column closed pattern subgroups U of the finite upper unitriangular groups U n (q) are defined as sets of matrices in U n (q) having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation [9] and apply this to CU yielding a generalisation of Yan's coadjoint cluster representations of [11] Then we give a complete classification of the resulting supercharacters, by describing the resulting orbits and determining the Hom-spaces between orbit modules.2.2 Corollary. Let G act on V as above. Then the number of orbits of the actions of G on V andV coincide.Proof. In view of 2.1 it suffices to show, that for all g ∈ G the number of elements v ∈ V satisfying v.g = v and τ ∈V satisfying τ.g = τ coincide. Now for g ∈ G and τ ∈V we have τ.g = τ if and only if τ.g −1 = τ . But this is equivalent toConsequently the number of linear characters of V fixed by g ∈ G is exactly the index [V :Obviously the maps v → v.g − v is an epimorphism from V onto W g and the kernel K of that map is the set of elements of V fixed by g ∈ G. But W g ∼ = V /K and hence |W g | = |V | |K| . This implies |K| = |V | |Wg| = [V : W g ] and the result follows.Note that given a map f : G → V we always have a map f * :for all x, g ∈ G.Positions in the triangle not in J are set to be zero as well.J from left and right by multiplication (denoted here as ".") and hence on V * J as well. By [4, Lemma 4.1] the numbers of U J -orbits (left, right, bi) on V J and V * J coincide. 3.11 Remark. We choose once for all a non trivial character θ : (F q , +) −→ C * . Obviously, for A ∈ V J , θ composed with the F q -linear map A * : V J → F q is a linear character of V J , consider the map (u, v) → utv from U J × U J to U J .t.U J . This map is clearly surjective, and it is easy to see that all elements of U J .t.U J are hit equally often. Each element of U J .t.U J , therefore, has the form u.t.v for exactly |U J | 2 |U J .t.U J | ordered pairs (u, v), and it follows that

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“…There has been considerable progress in recent years on the combinatorial representation theory of finite unipotent groups (some recent examples include [2,4,6,9]). For example, the representation theory of the maximal unipotent subgroup UT n (F q ) of the finite general linear group GL n (F q ) has developed from a wild problem to a combinatorial theory based on set partitions [3,18].…”

Section: Introductionmentioning

confidence: 99%

The combinatorics of GL n generalized Gelfand-Graev characters

Andrews

Thiem

2017

J. London Math. Soc.

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Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, generalized Gelfand-Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's definition in type A in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand-Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresponding generalized Gelfand-Graev characters in terms of unipotent representations, thereby recovering the Kostka-Foulkes polynomials as multiplicities.

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Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Cited by 8 publications

References 23 publications

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

Supercharacter theories of type $A$ unipotent radicals and unipotent polytopes

On monomial linearisation and supercharacters of pattern subgroups

The combinatorics of GL n generalized Gelfand-Graev characters

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