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Register a new userThe quadratic type of the 2-principal indecomposable modules of the double covers of alternating groups
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Authors
John Murray
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We determine the quadratic type of the 2-modular principal indecomposable modules of the double covers of alternating groups.
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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 dual modules of all irreducible modules of alternating groups over fields of characteristic 2.
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.
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 present sum-sides for principal characters of all standard (i.e., integrable and highest-weight) irreducible modules for the affine Lie algebra $A_2^{(2)}$. We use modifications of five known Bailey pairs; three of these are sufficient to obtain all the necessary principal characters. We then use the technique of Bailey lattice appropriately extended to include out-of-bounds values of one of the parameters, namely, $i$. We demonstrate how the sum-sides break into six families depending on the level of the modules modulo 6, confirming a conjecture of McLaughlin--Sills.
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