scholarly journals Twisted Dickson–Mui invariants and the Steinberg module multiplicity

2011 ◽  
Vol 151 (1) ◽  
pp. 43-57 ◽  
Author(s):  
JINKUI WAN ◽  
WEIQIANG WANG
Keyword(s):  
Tensor Product ◽  
Linear Group ◽  
General Linear Group ◽  
Exterior Algebra ◽  
Symmetric Algebra ◽  
General Linear ◽  
Parabolic Subgroups ◽  
Steinberg Module ◽  
Finite General Linear Group

AbstractWe determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S•(V) and the exterior algebra ∧•(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S•(V) ⊗ ∧•(V), twisted by an arbitrary determinant power.

scholarly journals THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS

2005 ◽  
Vol 92 (1) ◽  
pp. 62-98 ◽  
Author(s):  
BERND ACKERMANN
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Indecomposable Module ◽  
Defect Group ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Projective Indecomposable Module ◽  
Finite General Linear Group

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.

scholarly journals ASYMPTOTIC BOUNDS FOR THE SIZE OF Hom(A, GLn(q))

2017 ◽  
Vol 60 (1) ◽  
pp. 51-61
Author(s):  
MICHAEL BATE ◽  
ALEC GULLON
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Asymptotic Bounds ◽  
Arbitrary Finite Group ◽  
Finite General Linear Group

AbstractFix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

scholarly journals On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Tensor Product ◽  
Finite Field ◽  
Linear Group ◽  
General Linear Group ◽  
Polynomial Algebra ◽  
General Linear ◽  
Minimal Set ◽  
Equivalent Problem ◽  
Monomial Basis ◽  
Trivial Subgroup

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each $t$ and degree $n\geq 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\geq 5,$ even for the trivial subgroup $G_t = \{e\}.$ In the present paper, we consider the subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators of $(\mathbb P^{\otimes t})^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

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