scholarly journals p-Filtrations and the Steinberg Module

2001 ◽  
Vol 244 (2) ◽  
pp. 664-683 ◽  
Author(s):  
Henning Haahr Andersen
Keyword(s):  
Steinberg Module

The Steinberg module of a graph

1990 ◽  
Vol 55 (5) ◽  
pp. 503-506 ◽  
Author(s):  
Dina Ghinelli ◽  
Dieter Jungnickel
Keyword(s):  
Steinberg Module

scholarly journals Stability in the high-dimensional cohomology of congruence subgroups

2020 ◽  
Vol 156 (4) ◽  
pp. 822-861
Author(s):  
Jeremy Miller ◽  
Rohit Nagpal ◽  
Peter Patzt
Keyword(s):  
Congruence Subgroup ◽  
Stability Result ◽  
High Dimensional ◽  
Vanishing Theorem ◽  
Congruence Subgroups ◽  
Codimension One ◽  
Representation Stability ◽  
Steinberg Module ◽  
Finiteness Properties ◽  
Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

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.

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