## 2-groups with an odd-order automorphism that is the identity by Mazurov V.D.

By Mazurov V.D.

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Additional info for 2-groups with an odd-order automorphism that is the identity on involutions

Example text

19) normalized so that v∅ v∅ = 1. 3 Degenerate affine Hecke algebra In this chapter we define the degenerate affine Hecke algebra n . As a vector xn of the group algebra space, n is the tensor product FSn ⊗ F x1 xn . Moreover, FSn and the free commutative polynomial algebra F x1 xn are subalgebras of n isomorphic to FSn and FSn ⊗ 1 and 1 ⊗ F x1 xn , respectively. Furthermore, there exists an algebra homomorF x1 phism n → FSn , which is the “identity” on the subalgebra FSn , that is sends w ⊗1 to w, see Chapter 7.

As x and z act on wT by where g ∈ Sk , x ∈ Ln−k+1 multiplication with scalars, the result follows. 10 If / is not connected, then / 1 2 k = 0. Proof Let / = ∪ , where and are skew shapes disconnected from each other, that is res C − res D > 1 for any C ∈ and D ∈ . Let c = and d = . There exists an / -tableau T such that T1 Tc ∈ and Tk ∈ . 5, the subspace of V / , spanned by Tc+1 vectors wT for all such tableaux T , is invariant with respect to Sc × Sd < Sk , V . 9 and and, as a Sc ×Sd -module, it is isomorphic to V V → Frobenius reciprocity, we get a surjective homomorphism indSk V V / .

Module. 2 (“Mackey Theorem”) Let M be an resn indn M admits a filtration with subquotients isomorphic to ind ∩x x resx−1 ∩ M one for each x ∈ D . Moreover, the subquotients can be taken in any order refining the Bruhat order on D , in particular ind ∩ res ∩ M appears as a submodule. 1 and the isomorphism ⊗ x ∩x which is easy to check. 17) for all i = 1 n−1 j = 1 n. If M is a finite dimensional n -module, we can use to make the dual space M ∗ into an n -module denoted M . Note leaves invariant every parabolic subalgebra of n , so also induces a duality on finite dimensional -modules for each composition of n.