Leonard Scott
Growth rates and complexity for quantum group cohomology with applications to Kazhdan-Lusztig polynomials.
Mon, 04/26/2010 - 15:01 — SomNambulAuthors: Brian Parshall, Leonard Scott
Subjects: Representation Theory
AbstractIn previous work, the authors established various bounds for the dimensions
of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of
semisimple algebraic groups (in positive characteristic) and quantum groups (at
roots of unity). Generally, these bounds depend only on the root system (and
not on the characteristic $p$ or the size of the root of unity). This paper
investigates the rate of growth of these bounds.Integral and graded quasi-hereditary algebras, II with applications to representations of generalized $q$-Schur algebras and algebraic groups.
Tue, 10/06/2009 - 17:01 — SomNambulAuthors: Brian Parshall, Leonard Scott
Subjects: Group Theory
AbstractWe consider a pair $(\fa,A)$ consisting of a quasi-hereditary algebra $A$ and
a (positively) graded subalgebra $\mathfrak a$. We present conditions which
guarantee that the algebra $\gr A$ obtained by grading $A$ by its radical
filtration is quasi-hereditary and Koszul. In such cases, we also show that the
standard and costandard modules for $A$ have a structure as graded modules for
$\fa$. These results are applied to obtain new information about the finite
dimensional algebras (e. g., the $q$-Schur algebras) which arise from quantum
enveloping algebras.Bounding Ext for modules for algebraic groups, finite groups, and quantum groups.
Thu, 10/01/2009 - 00:35 — SomNambulAuthors: Brian Parshall, Leonard Scott
Subjects: Group Theory
AbstractGiven a finite root system $\Phi$, we show there is an integer $c=c(\Phi)$
such that $\dim\Ext_G^1(L,L')<c$, for any reductive algebraic group $G$ with
root system $\Phi$ and any irreducible rational $G$-modules $L,L'$. We also
prove that there is such a bound in the case of finite groups of Lie type,
depending only on the root system and not on the underlying field. For quantum
groups, we are able to obtain a similar result for $\Ext^n$, for any integer
$n\geq 0$, using a constant depending only on $n$ and the root system.
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