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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 $s,tin G$ 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 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, $G$-invariant form which do not decompose into smaller such modules, and clas
One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation $R$-lattice for the finite $p$-group $G$ in terms of the restriction to a normal subgroup $N$ and the $N$-fix
Let $alpha$ be a composition of $n$ and $sigma$ a permutation in $mathfrak{S}_{ell(alpha)}$. This paper concerns the projective covers of $H_n(0)$-modules $mathcal{V}_alpha$, $X_alpha$ and $mathbf{S}^sigma_{alpha}$, which categorify the dual immacula
We construct, for any finite commutative ring $R$, a family of representations of the general linear group $mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $mathrm{GL}_n$ over a finite field.