Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuadratic principal indecomposable modules and strongly real elements of finite Groups
110
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $G$.
rate research
Read More
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 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.
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 $N$-fixed points of the lattice, where $R$ is a finite extension of the $p$-adic integers. Using techniques from relative homological algebra, we generalize Weiss Theorem to the class of infinitely generated pseudocompact lattices for a finite $p$-group, allowing $R$ to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.
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 immaculate quasisymmetric function, the extended Schur function, and the quasisymmetric Schur function when $sigma$ is the identity, respectively. First, we show that the projective cover of $mathcal{V}_alpha$ is the projective indecomposable module $mathbf{P}_alpha$ due to Norton, and $X_alpha$ and the $phi$-twist of the canonical submodule $mathbf{S}^{sigma}_{beta,C}$ of $mathbf{S}^sigma_{beta}$ for $(beta,sigma)$s satisfying suitable conditions appear as $H_n(0)$-homomorphic images of $mathcal{V}_alpha$. Second, we introduce a combinatorial model for the $phi$-twist of $mathbf{S}^sigma_{alpha}$ and derive a series of surjections starting from $mathbf{P}_alpha$ to the $phi$-twist of $mathbf{S}^{mathrm{id}}_{alpha,C}$. Finally, we construct the projective cover of every indecomposable direct summand $mathbf{S}^sigma_{alpha, E}$ of $mathbf{S}^sigma_{alpha}$. As a byproduct, we give a characterization of triples $(sigma, alpha, E)$ such that the projective cover of $mathbf{S}^sigma_{alpha, E}$ is indecomposable.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
