U(q) acting on flags and supercharacters

Abstract

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.

On monomial linearisation and supercharacters of pattern subgroups

“…As we shall see there is a local "mainhook-wise" version of formula 5.5 which multiplied up over all main hooks gives the global formula 5.5. 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).…”

“…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.…”

“…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 rà and by the free choices of λ and ρ, we have q 2 many different right orbits shown to be isomorphic.…”

“…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…”

“…the idempotent of CV J , associated with the complex conjugate character [A] which is obviously [−A], since V J is written additively. In [6] we called e A = q −|J| [−A] ∈ CV J hence lidempotent, to distinguish those from idempotents of the group algebra CU J and since it is really an idempotent for the additive group of the Lie algebra Lie(U J ) = V J . Moreover, for simplicity we call the linear characters [A] in V "lidempotents" as well, although [A] is the multiple q |J| e −A of the idempotent e −A in the group algebra C V = CV .…”