Let be an odd prime and let B be p-block of a finite group, such that B has cyclic defect groups. We describe the self-dual indecomposable B-modules and for each such module determine whether it is symplectic or orthogonal.

Let G be a finite group, let N be a normal subgroup of G and let B be an irreducible character of N. P.X. Gallagher showed that the number of irreducible characters of G lying B equals the number of so-called B-good conjugacy classes of G & theta;. Here we count the number of real irreducible characters of G lying over B. To do so, we assign a new invariant, with value 0, +1 or -1, to each good conjugacy class. Then the desired number is the sum over these invariants. We also compute the Frobenius-Schur indicator of the induced character B & UARR;G using a similar formula.& COPY; 2023 The Author(s). Published by Elsevier Inc.This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4.0/).

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius–Schur indicator 1 can also be expressed in group theoretical terms. We show that this can done by counting solutions of g12…gn2=1 with g1,…,gn∈G.

We prove Clifford theoretic results on the representations of finite groups which only hold in characteristic $2$. Let $G$ be a finite group, let $N$ be a normal subgroup of $G$ and let $\varphi$ be an irreducible $2$-Brauer character of $N$ which is self-dual. We prove that there is a unique self-dual irreducible Brauer character $\theta$ of $G$ such that $\varphi$ occurs with odd multiplicity in the restriction of $\theta$ to $N$. Moreover this multiplicity is $1$. Conversely if $\theta$ is an irreducible $2$-Brauer character of $G$ which is self-dual but not of quadratic type, the restriction of $\theta$ to $N$ is a sum of distinct self-dual irreducible Brauer character of $N$, none of which have quadratic type. Let $b$ be a real $2$-block of $N$. We show that there is a unique real $2$-block of $G$ covering $b$ which is weakly regular.

For each positive integer n, we construct a bijection between the odd partitions of n and the distinct partitions of n. Our bijection extends a bijection of Bressoud between the odd-and-distinct partitions of n and the splitting partitions of n. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with one recently constructed by Chen, Gao, Ji and Li.

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Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.

Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic form if and only if there are involutions $s,t\in G$ such that $st$ has odd order and $\varphi(st)/2$ is not an algebraic integer. We then show that the number of isomorphism classes of quadratic principal indecomposable $G$-modules is equal to the number of strongly real conjugacy classes of odd order elements of $G$.

We determine all indecomposable symplectic modules for the Klein-four group K-4 over a perfect field of characteristic 2 and classify the symplectic forms up to isometry. We also determine all K-4-invariant quadratic forms which polarize to a given symplectic form and classify such quadratic forms up to isometry. (C) 2018 Elsevier Inc. All rights reserved.

Let G be a finite solvable or symmetric group, and let B be a 2-block of G. We construct a canonical correspondence between the irreducible characters of height zero in B and those in its Brauer first main correspondent. For symmetric groups our bijection is compatible with restriction of characters.

We determine the dual modules of all irreducible modules of alternating groups over fields of characteristic 2.

We determine the quadratic type of the 2-modular principal indecomposable modules of the double covers of alternating groups.

For the Klein-Four Group G and a perfect field k of characteristic two we determine all indecomposable symplectic kG-modules, that is, kG-modules with a symplectic, Ginvariant form which do not decompose into smaller such modules, and classify them up to isometry. Also we determine all quadratic forms that have one of the above symplectic forms as their associated bilinear form and describe their isometry classes.

For the Klein-Four Group $G$ and a perfect field $k$ of characteristic two we determine all indecomposable symplectic $kG$-modules, that is, $kG$-modules with a symplectic, $G$-invariant form which do not decompose into smaller such modules, and classify them up to isometry. Also we determine all quadratic forms that have one of the above symplectic forms as their associated bilinear form and describe their isometry classes.

We prove a number of results about the ordinary and Brauer characters of finite solvable groups in characteristic 2, by defining and using the concept of the extended nucleus of a real irreducible character. In particular we show that the Isaacs canonical lift of a real irreducible Brauer character has Frobenius–Schur indicator +1. We also show that the principal indecomposable module corresponding to a real irreducible Brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus.

Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $2$ and let $M$ be an indecomposable $kG$-module which affords a non-degenerate $G$-invariant symmetric bilinear form. We introduce the symmetric vertices of $M$. Each of these is a $2$-subgroup of $G$ which contains a Green vertex of $M$ with index at most $2$. If $M$ is irreducible then its symmetric vertices are determined up to $G$-conjugacy. If $B$ is the real $2$-block of $G$ containing $M$, we show that each symmetric vertex of $M$ is contained in an extended defect group of $B$. Moreover, we characterise the extended defect groups in terms of symmetric vertices. In order to prove these results, we develop the theory of involutary $G$-algebras. This allows us to translate questions about symmetric $kG$-modules into questions about projective modules of quadratic type.

M. Kiyota, T. Okuyama and T. Wada recently proved that each 2-block of a finite symmetric group contains a unique irreducible Brauer character that has height 0. We present a more conceptual proof of this result.

Let (F, R, k) be a p-modular system, and let denote the centralizer of the symmetric group S ℓ in the group algebra RS n , where ℓ ≤n. We show that the decomposition map of can be determined from that of the degenerate affine Hecke algebra of rank n − ℓ. We use this to determine the blocks of for ℓ =n − 2, n − 3. For each p-core κ, there is an n 0 such that if n > n 0 and E n is a block idempotent of RS n with core κ, then E n E n−ℓ is zero or a block idempotent of , for each block idempotent E n−ℓ of RS n−ℓ.

We give the lower bounds 2ℓ and 2ℓ−2+2ℓ−4+⋯ for the number of irreducible characters of height zero and positive height, respectively, in a p-block of a p-solvable group, where ℓ is the p-length of an associated p-solvable group. We also prove some results on extensions of linear characters in p-subgroups of p-solvable groups.