Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system is a deformation of the group algebra of defined as follows. Take the free -module with basis , and impose the multiplicative relations if , and otherwise. (For now, ignore the square root of .) Humphreys proves that these relations describe a unique associative algebra structure on [...]

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Chevalley-Bruhat order

Before we define Bruhat order, I’d like to say a few things by way of motivation. Warning: I know nothing about algebraic groups, so take everything I say with a grain of salt. A (maximal) flag in a vector space of dimension is a chain of subspaces such that . The flag variety of is, [...]

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The strong exchange condition

It’s nice that Weyl groups are Coxeter groups and all, but the definition of a Coxeter group as a group with a particular kind of representation doesn’t immediately tell us why this is the appropriate level of generalization (although the faithfulness of the geometric representation is a good sign). It turns out there is a [...]

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Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of [...]

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