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Register a new userModules of the $0$-Hecke algebra arising from standard permuted composition tableaux
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We study the $H_n(0)$-module $mathbf{S}^sigma_alpha$ due to Tewari and van Willigenburg, which was constructed using new combinatorial objects called standard permuted composition tableaux and decomposed into cyclic submodules. First, we show that every direct summand appearing in their decomposition is indecomposable and characterize when $mathbf{S}^sigma_alpha$ is indecomposable. Second, we find characteristic relations among $mathbf{S}^sigma_alpha$s and expand the image of $mathbf{S}^sigma_alpha$ under the quasi characteristic in terms of quasisymmetric Schur functions. Finally, we show that the canonical submodule of $mathbf{S}^sigma_alpha$ appears as a homomorphic image of a projective indecomposable module.
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The purpose of this paper is to provide a unified method for dealing with various 0-Hecke modules constructed using tableaux so far. To do this, we assign a $0$-Hecke module to each left weak Bruhat interval, called a weak Bruhat interval module. We prove that every indecomposable summand of the $0$-Hecke modules categorifying dual immaculate quasisymmetric functions, extended Schur functions, quasisymmetric Schur functions, and Young row-strict quasisymmetric Schur functions is a weak Bruhat interval module. We further study embedding into the regular representation, induction product, restriction, and (anti-)involution twists of weak Bruhat interval modules.
We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of the image of spherical and Iwahori-fixed vectors in the unramified principal series for this class of models. We provide an explicit alternator expressionfor the image of the spherical vectors under these functionals in terms of the representation theory of the dual group.
We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on $p$-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplect
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.
In a previous work (arXiv:0806.1503v2), we defined a family of subcomplexes of the $n$-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least $k$, and we proved that the homology of such a subcomplex is concentrated in degree $k-1$. This homology group supports a natural action of the Coxeter group $W(D_n)$ of type $D$. In this paper, we explicitly determine the characters (over ${Bbb C}$) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group $S_n$ by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of $sym_n$ agree (over ${Bbb C}$) with the representations of $sym_n$ on the $(k-2)$-nd homology of the complement of the $k$-equal real hyperplane arrangement.
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