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

Andrews, Scott

doi:10.48550/arxiv.1502.06542

Cited by 1 publication

(1 citation statement)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here M (λ) denotes the permutation representation of GL n (q) acting by right translation on the set of all λ-flags in the natural GL n (q)-module V = F n q , where F q is the field with q elements. In a recent paper, Andrews [14] gave an alternate construction of the unipotent Specht modules based on generalized Gelfand-Graev characters. If K = C then S(λ) are irreducible unipotent GL n (q)-modules appearing as irreducible constituents of the permutation module of GL n (q) acting on flags in V .…”

Section: Introductionmentioning

confidence: 99%

U(q) acting on flags and supercharacters

Dipper

Guo

2017

Journal of Algebra

View full text Add to dashboard Cite

Let U = U n (q) be the group of lower unitriangular n× n-matrices with entries in the field F q with q elements for some prime power q and n ∈ N. We investigate the restriction to U of the permutation action of GL n (q) on flags in the natural GL n (q)-module F n q . Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of GL n (q) on the cosets of maximal parabolic subgroups into irreducible CU -modules. the following lemma is seen immediately:2.2 Lemma. Let G be a finite group, H G and let e = h∈H h and M = eK G . Then M consists of functions from G to K, which are constant on the right cosets of H in G. Now let G be a finite group acting on a finite abelian group (V, +) by automorphisms, the actionObserve the group algebra CV ∼ = C V of V becomes a right CG-module settingwhere the action of G on C V is denoted by (τ, g) → τ.g. We denoteV to be the set of linear characters of V , i.e. that is of group homomorphisms of (V, +) into to the multiplicative group C * = C \ {0} of C. Note thatV is contained in C V . Indeed under the identification of C V and CV , χ ∈V is mapped to |V |eχ whereχ is the complex conjugate character of V and eχ is the primitive idempotent of CV affordingχ. Since {e χ | χ ∈V } is a basis of CV , we conclude thatV is a C-basis of C V . Moreover one checks immediately:2.4 Lemma. The action (τ, g) → τ.g of G on C V permutesV . So, for χ ∈V , g ∈ G we have χ.g ∈V . Now let f : G → V be a 1-cocycle. Then Jedlitschky showed in [12]:2.5 Theorem. The group algebra C V becomes a monomial CG-module with monomial basiŝ V by setting χg = χ(f (g −1 ))χ.g for χ ∈V , g ∈ G.Note that for 1-cocycle f : G → V (left or right) we always have f (1 G ) = 0 ∈ V . 2.6 Lemma. Let f : G → V be a 1-cocycle (or left 1-cocycle). Then ker f = {g ∈ G | f (g) = 0 ∈ V } is a subgroup of G. Proof. Let a, b ∈ ker f , then f (ab) = f (a)b + f (b) = 0 · b + 0 = 0 proving ab ∈ ker f . Moreover 0 = f (1) = f (aa −1 ) = f (a)a −1 + f (a −1 ) = f (a −1 )showing a −1 ∈ ker f . The case of a left 1-cocycle f is shown similarly.

show abstract

Section: Introductionmentioning

confidence: 99%