Hecke algebras | Annoying Precision

Posts Tagged ‘Hecke algebras’

The double commutant theorem

Posted in module theory, tagged adjoint functors, Hecke algebras, MaBloWriMo on November 11, 2012 | 3 Comments »

Let $A$ be an abelian group and $T = \{ T_i : A \to A \}$ be a collection of endomorphisms of $A$ . The commutant $T'$ of $T$ is the set of all endomorphisms of $A$ commuting with every element of $T$ ; symbolically,

$\displaystyle T' = \{ S \in \text{End}(A) : TS = ST \}$ .

The commutant of $T$ is equal to the commutant of the subring of $\text{End}(A)$ generated by the $T_i$ , so we may assume without loss of generality that $T$ is already such a subring. In that case, $T'$ is just the ring of endomorphisms of $A$ as a left $T$ -module. The use of the term commutant instead can be thought of as emphasizing the role of $A$ and de-emphasizing the role of $T$ .

The assignment $T \mapsto T'$ is a contravariant Galois connection on the lattice of subsets of $\text{End}(A)$ , so the double commutant $T \mapsto T''$ may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

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Hecke algebras and the Kazhdan-Lusztig polynomials

Posted in algebraic combinatorics, representation theory, tagged Coxeter groups, duality, finite fields, Hecke algebras, Kazhdan-Lusztig, q-analogues on July 12, 2010 | 2 Comments »

The Hecke algebra attached to a Coxeter system $(W, S)$ is a deformation of the group algebra of $W$ defined as follows. Take the free $\mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]$ -module $\mathcal{H}_W$ with basis $T_w, w \in W$ , and impose the multiplicative relations

$T_w T_s = T_{ws}$

if $\ell(sw) > \ell(w)$ , and

$T_w T_s = q T_{ws} + (q - 1) T_w$

otherwise. (For now, ignore the square root of $q$ .) Humphreys proves that these relations describe a unique associative algebra structure on $\mathcal{H}_W$ with $T_e$ as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When $W$ is the Weyl group of an algebraic group $G$ with Borel subgroup $B$ , the above relations describe the algebra of functions on $G(\mathbb{F}_q)$ which are bi-invariant with respect to the left and right actions of $B(\mathbb{F}_q)$ under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group $G$ , and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing $G$ and $B$ with $\text{SL}_2(\mathbb{Q})$ and $\text{SL}_2(\mathbb{Z})$ gives the Hecke operators in the theory of modular forms.

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Annoying Precision

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Posts Tagged ‘Hecke algebras’

The double commutant theorem

Hecke algebras and the Kazhdan-Lusztig polynomials

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