Representations of the double Burnside algebra and cohomology of the extraspecial p -group

Abstract

Let E be the extraspecial p-group of order p 3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence. It is well known that indecomposable summands in the complete stable splitting correspond to simple modules for the mod p double Burnside algebra. We shall use representation theory of the double Burnside algebra and the theory of biset functors. X ij 1991 Mathematics Subject Classification. Primary 55P35, 57T25, 20C20; Secondary 55R35, 57T05.Let Λ be a finite dimensional algebra over a filed k. We denote the Jacobson radical of Λ by rad(Λ), namely, rad(Λ) is the intersection of all maximal ideals of Λ. If e is a primitive idempotent in Λ, then eΛ is a projective indecomposable right Λ-module and eΛ/erad(Λ) is a simple right Λ-module. Let 1 = 1≤i≤l 1≤j≤m(i) e ij be a decomposition of unity into primitive orthogonal idempotents in Λ, where e ij Λ/e ij rad(Λ) ≃ e mn Λ/e mn rad(Λ) if and only of i = m. Thengives the complete set of representatives of isomorphism classes of simple right Λ-modules. Let e i = 1≤j≤m(i) e ij and, in this paper, we call e i an idempotent corresponding to the simple module S i . Multiplication by e i induces the identity map on S i and S j e i = 0 for j = i. On the other hand, we have S i e i1 ≃ Hom Λ (e i1 Λ, S i ) ≃ End Λ (S i ) and the multiplicity m(i) is equal to dim End Λ (Si) S i .Let M be a finite dimensional right Λ-module. Then M has a composition seriesIf every composition factor of M is isomorphic to S i , then M = M e i and the right multiplication by e i induces the identity on M . In general case, to determine M e i , we shall use the following lemma in section 10. Note that H * (E) is not finite dimensional, but each homogeneous part H n (E) is a finite dimensional A p (E, E)module and we can apply the lemma. Lemma 2.1. Let M be a finite dimensional Λ-module and S a simple Λ-module. Let e be an idempotent corresponding to S. Let 0 ⊂ L ⊂ N ⊂ M

Representations of finite sets and correspondences

“…(Actually, as observed by Hida and Yagita in Lemma 3.1 of [HY14], we have an equality F ′ 1 = F 1 , because both subfunctors are generated by their common evaluation at Y . )…”

Representations of the double Burnside algebra and cohomology of the extraspecial p-group II

“…In particular, their result implies the classification of simple A p (E, E)-modules. [7,Proposition 10.1]). The simple A p (E, E)-modules are given as follows:…”

“…We consider the multiplicity m(G, 2) m and m(G, 2) 2m in some cases. [7,Corollary 9.3]. This module has a basis…”

“…Let p be an odd prime and E = p 1+2 + the extraspecial p-group of order p 3 and exponent p. In the previous paper [7], we determined the composition factor of H * (E) = (F p ⊗ H * (E, Z))/ (0) as a right A p (E, E)-module, where A p (E, E) is a double Burnside algebra of E over F p = Z/pZ. In this paper, we consider the whole part of the cohomology H * (E, F p ) and determine the composition factor of H * (E, F p ) as an A p (E, E)-module.…”

“…G -rad = {A 0 , A ∞ } and Out F G (E) = 3I, u = H. On the other hand, if we take G -rad = {A 1 , A 6 } and Out F G (E) = 3I, w .The inclusions of fusion systems are obtained by the table above. For F i ′ 24 ←− L 3 (7) and F i 24 ←− L 3 (7) : 2, we use the second rows of L 3(7) and L 3 (7) : 2. The information on the summands are obtained by Example 4.21 and Theorem 4.22.…”