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Register a new userIrreducible representations of the symmetric groups from slash homologies of p-complexes
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In the 40s, Mayer introduced a construction of (simplicial) $p$-complex by using the unsigned boundary map and taking coefficients of chains modulo $p$. We look at such a $p$-complex associated to an $(n-1)$-simplex; in which case, this is also a $p$-complex of representations of the symmetric group of rank $n$ - specifically, of permutation modules associated to two-row compositions. In this article, we calculate the so-called slash homology - a homology theory introduced by Khovanov and Qi - of such a $p$-complex. We show that every non-trivial slash homology group appears as an irreducible representation associated to two-row partitions, and how this calculation leads to a basis of these irreducible representations given by the so-called $p$-standard tableaux.
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Answering a question of I. M. Isaacs, we show that the largest degree of irreducible complex representations of any finite non-abelian simple group can be bounded in terms of the smaller degrees. We also study the asymptotic behavior of this largest degree for finite groups of Lie type. Moreover, we show that for groups of Lie type, the Steinberg character has largest degree among all unipotent characters.
Let $G$ be an almost linear Nash group, namely, a Nash group which admits a Nash homomorphism with finite kernel to some $GL_k(mathbb R)$. A homology theory (the Schwartz homology) is established for the category of smooth Fre representations of $G$ of moderate growth. Frobenius reciprocity and Shapiros lemma are proved in this category. As an application, we give a criterion for automatic extensions of Schwartz homologies of Schwartz sections of a tempered $G$-vector bundle.
In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{Gamma}$ where $Gamma$ is a product of general linear groups over a field $K$ of characteristic zero, and $U$ is a finite dimensional rational representation of $Gamma$. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where $U=text{End}(W^{oplus k})$ and $Gamma = text{GL}(W)$ and on the case where $U=text{End}(Votimes W)$ and $Gamma = text{GL}(V)times text{GL}(W)$, though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.
We classify all triples $(G,V,H)$ such that $SL_n(q)leq Gleq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $Vdar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
We determine the multiplicities of irreducible summands in the symmetric and the exterior squares of hook representations of symmetric groups over an algebraically closed field of characteristic zero.
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