Posts Tagged ‘Hecke algebras’

Continuing yesterday’s story about relative positions, let G be a finite group and let X and Y be finite G-sets. Yesterday we showed that G-orbits on X \times Y can be thought of as “atomic relative positions” of “X-figures” and “Y-figures” in some geometry with symmetry group G, and further that if X \cong G/H and Y \cong G/K are transitive G-sets then these can be identified with double cosets H \backslash G / K.

Representation theory provides another interpretation of G-orbits on X \times Y as follows. First, if \mathbb{C}[X] is any permutation representation, then the G-fixed points \mathbb{C}[X]^G have a natural basis given by summing over G-orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations \mathbb{C}[X], \mathbb{C}[Y]. Because \mathbb{C}[X] is self-dual, we have

\displaystyle \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) \cong (\mathbb{C}[X] \otimes \mathbb{C}[Y])^G \cong \mathbb{C}[X \times Y]^G

and hence \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) has a natural basis given by summing over G-orbits of the action on X \times Y.

Definition: The G-morphism \mathbb{C}[X] \to \mathbb{C}[Y] associated to a G-orbit of X \times Y via the above isomorphisms is the Hecke operator associated to the G-orbit (relative position, double coset).

Below the fold we’ll write down some details about how this works and see how we can use the idea that G-morphisms between permutations have a basis given by Hecke operators to work out, quickly and cleanly, how some permutation representations decompose into irreducibles. At the end we’ll state another puzzle.


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The goal of this post is to explain something that the cool kids all understood ages ago (David Speyer, John Baez) but that I hadn’t internalized until recently.

Let G be a group and let X and Y be transitive G-sets, so X = G/H and Y = G/K for some subgroups H, K of G. In “geometric” situations (in the sense of the Erlangen program), G is the symmetry group of some kind of geometry (for example, affine geometry, or Euclidean geometry), and X and Y are spaces of “figures” in the geometry (for example, points, lines, or triangles). We’ll call the points of XX-figures” and similarly for Y.

Now, figures in a geometry can be in various “relative positions” (or “incidence relations”) with respect to each other: for example, a point can be contained in a line, or two lines can intersect at right angles. What makes these geometrically meaningful is that they are invariant under the symmetry group G of the geometry: for example, the condition that a point is contained in a line is invariant under affine symmetries, and the condition that two lines intersect at right angles is invariant under Euclidean symmetries. This motivates the following.

Definition: A relative position of X-figures and Y-figures is a G-invariant subset of X \times Y, or equivalently a G-invariant relation R : X \to Y.

Any G-invariant subset of X \times Y decomposes into a disjoint union of G-orbits: these are the atomic relative positions.

Proposition: G-orbits of the action of G on G/H \times G/K (equivalently, the atomic relative positions of X-figures and Y-figures) can canonically be identified with double cosets H \backslash G/K, via the map

\displaystyle G/H \times G/K \ni ([g_1], [g_2]) \mapsto [g_1^{-1} g_2] \in H \backslash G/K

where [g] \in G/H means the image of g \in G under G \to G/H.

This is the conceptual interpretation of double cosets. It took an annoyingly long time between the first time I was introduced to double cosets (which I believe was in 2010) and the time I internalized the above fact (which was this year, 2015). Unlike the usual definition, this interpretation naturally generalizes to a notion of “triple cosets” (G-orbits on a triple product X \times Y \times Z), and so forth.

Example. Let G = \text{Isom}(\mathbb{R}^n) be the group of isometries of Euclidean space, which more explicitly is the semidirect product \mathbb{R}^n \rtimes O(n). If X = Y are both the G-space of points in \mathbb{R}^n, then the atomic relative positions have the form “a point has distance r from another point,” where r is any nonnegative real.

Example. Let G = GL_n(k) be the general linear group over a field k and let H = K = B be the Borel subgroup of upper triangular matrices. G/B is the space of complete flags in V = k^n. As it turns out, there are exactly n! atomic relative positions of a pair of complete flags. When k is a finite field these form a basis of a Hecke algebra. In general they label the Bruhat decomposition of G.

For example, when n = 2, a complete flag is just a line in V = k^2, and there are two atomic relative positions: the lines can be identical or they can be different. When n = 3, a complete flag is a line V_1 contained in a plane V_2 in V = k^3, and there are six atomic relative positions. Letting W_1 \subset W_2 denote a second complete flag, they are

  • V_1 = W_1, V_2 = W_2,
  • V_1 = W_1, V_2 \neq W_2,
  • V_1 \neq W_1, V_2 = W_2,
  • V_1 \neq W_1, V_1 \subset W_2, W_1 \not\subset V_2, V_2 \neq W_2,
  • V_1 \neq W_1, W_1 \subset V_2, V_1 \not\subset W_2, V_2 \neq W_2,
  • V_1 \neq W_2, V_1 \not\subset W_2, W_1 \not\subset V_2, V_2 \neq W_2.

Instead of thinking about relations as conditions on a pair of complete flags, we can also think about them as partial multi-valued functions from complete flags to complete flags. In those terms the six atomic relative positions are

  • Do nothing,
  • Pick a different plane,
  • Pick a different line,
  • Pick a different plane still containing the original line, then pick a different line not contained in the original plane,
  • Pick a different plane not containing the original line, then pick a different line contained in the original plane,
  • Pick a different plane not containing the original line, then pick a different line not contained in the original plane.

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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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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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