scholarly journals The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials

2019 ◽  
Vol 23 (5) ◽  
pp. 188-204
Author(s):  
Martina Lanini ◽  
Arun Ram
Keyword(s):  
Tensor Product ◽  
Product Theorem ◽  
Llt Polynomials

scholarly journals HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

2020 ◽  
pp. 1-24
Author(s):  
MATTHEW WESTAWAY
Keyword(s):  
Tensor Product ◽  
Algebraic Groups ◽  
Preceding Paper ◽  
Irreducible Representations ◽  
Product Theorem ◽  
Enveloping Algebras ◽  
Frobenius Kernel ◽  
Frobenius Kernels ◽  
Lie Algebra Representations ◽  
Semisimple Algebraic Groups

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

The tensor product theorem for &∇tilde;-nilpotence and the dimension of unstable modules

2001 ◽  
Vol 130 (3) ◽  
pp. 427-439
Author(s):  
GEOFFREY M. L. POWELL
Keyword(s):  
Tensor Product ◽  
Steenrod Algebra ◽  
Transcendence Degree ◽  
Vector Spaces ◽  
Product Theorem ◽  
Finite Dimensional ◽  
Noetherian Algebra

Let [Fscr ] be the category of functors from the category of finite-dimensional [ ]2-vector spaces to [ ]2-vector spaces. The concept of &∇tilde;-nilpotence in the category [Fscr ] is used to define a ‘dimension’ for the category of analytic functors which has good properties. In particular, the paper shows that the tensor product F [otimes ] G of analytic functors which are respectively &∇tilde;s and &∇tilde;t nilpotent is &∇tilde;s+t − 1-nilpotent.The notion of &∇tilde;-nilpotence is extended to define a dimension in the category of unstable modules over the mod 2 Steenrod algebra, which is shown to coincide with the transcendence degree of an unstable Noetherian algebra over the Steenrod algebra.

scholarly journals Tensor product theorem for Hitchin pairs – An algebraic approach

2011 ◽  
Vol 61 (6) ◽  
pp. 2361-2403 ◽  
Author(s):  
V. Balaji ◽  
A.J. Parameswaran
Keyword(s):  
Tensor Product ◽  
Algebraic Approach ◽  
Product Theorem

scholarly journals A tensor product theorem related to perfect crystals

2003 ◽  
Vol 267 (1) ◽  
pp. 212-245 ◽  
Author(s):  
Masato Okado ◽  
Anne Schilling ◽  
Mark Shimozono
Keyword(s):  
Tensor Product ◽  
Product Theorem

Complexity and varieties for infinitely generated modules, II

1996 ◽  
Vol 120 (4) ◽  
pp. 597-615 ◽  
Author(s):  
D. J. Benson ◽  
Jon F. Carlson ◽  
J. Rickard
Keyword(s):  
Tensor Product ◽  
Matrix Representation ◽  
Cohomology Ring ◽  
Group Cohomology ◽  
Product Theorem ◽  
The Matrix ◽  
A Finite Group ◽  
Infinitely Generated Modules ◽  
Rank Variety ◽  
Associated Variety

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

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