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## Hecke operators are also relative positions

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.

## Double cosets are relative positions

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 $X$$X$-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.

## The double commutant theorem

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.

## Hecke algebras and the Kazhdan-Lusztig polynomials

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.