Posted by: Steven Sam | October 12, 2009

GLFq III: characteristic map

In the last post of this series, I gave some definitions and facts regarding the Hall–Littlewood functions. I also sketched the relationship between symmetric functions and representations of the symmetric group. Now we’ll see how this works for the finite general linear groups.

We want to imitate the Frobenius character that is used to relate the characters of the symmetric group to the ring of symmetric functions. But since the description of the conjugacy classes of the finite general linear group (and hence the parametrization of its irreducible characters) are more complicated than the description for the symmetric group, we’ll need a bigger ring to work with.

We continue the notation from the first post. For each irreducible polynomial $f \in \Phi$ and each positive integer i>0, we have a variable $X_{i,f}$, whose degree we set to be deg(f). For any symmetric function u, we set $u(X_f) = u(X_{1,f}, X_{2,f}, \dots)$. The graded ring we work in is $B = {\bf C}[e_n(X_f) \mid n \ge 1,\ f \in \Phi]$, where $e_n$ denotes the elementary symmetric function. In other words, elements of B are functions which are symmetric in each family of variables $X_f$.

Recall from last time that for a partition $\lambda$, $P_\lambda(x;t)$ and $Q_\lambda(x;t)$ are the Hall–Littlewood and augmented Hall–Littlewood functions. We use these to define elements in B:

$\tilde{P}_\lambda(X_f) = q^{-\deg(f) n(\lambda)} P_\lambda(X_f; q^{-\deg(f)})$

$\tilde{Q}_\lambda(X_f) = q^{\deg(f)(|\lambda| + n(\lambda))} Q_\lambda(X_f; q^{-\deg(f)})$,

where $n(\lambda) = \sum_i (i-1)\lambda_i$. For a partition-valued function $\boldsymbol{\mu}$, we set

$\displaystyle \tilde{P}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{P}_{\boldsymbol{\mu}(f)}(X_f)$,

$\displaystyle \tilde{Q}_{\boldsymbol{\mu}} = \prod_{f \in \Phi} \tilde{Q}_{\boldsymbol{\mu}(f)}(X_f)$.

We use these two bases to define a (complex) inner product on B:

$\langle \tilde{P}_{\boldsymbol{\lambda}}, \tilde{Q}_{\boldsymbol{\mu}} \rangle = \delta_{\boldsymbol{\lambda}, \boldsymbol{\mu}}$.

Now we need to construct the representation ring of the finite general linear groups $G_n$ over ${\bf F}_q$, where now q will remain fixed. This will be very similar to what happens for the symmetric groups. Given characters u and v for $G_n$ and $G_m$, respectively, let P be the parabolic subgroup of $G_{n+m}$ consisting of matrices of the form

$g(A,B,C) = \begin{bmatrix} A & B \\ 0 & C \end{bmatrix}$

where A is an $n \times n$ matrix, B is an $n \times m$ matrix, and C is an $m \times m$ matrix. We define a character w on P by setting

$w(g(A,B,C)) = u(A) v(C)$.

Then the induction product $u \circ v$ is defined as the induced character ${\rm Ind}_P^{G_{n+m}}(w)$. (Recall that for symmetric groups, we define the induction product by inducing from parabolic subgroups $\mathfrak{S}_n \times \mathfrak{S}_m \subset \mathfrak{S}_{n+m}$.) If we let $A_n$ denote the complex vector space of characters of $G_n$, then the induction product gives a graded ring structure on $A = \bigoplus_{n \ge 0} A_n$. We can also put a complex inner product on A by setting the different graded components to be mutually orthogonal and using the standard inner product for characters on each component, just as in the case of the symmetric group. Now comes the important part: let $\pi_{\boldsymbol{\mu}}$ denote the function which is 1 on the conjugacy class corresponding to $\boldsymbol{\mu}$, and 0 elsewhere. Then we have a characteristic map

${\rm ch} \colon A \to B, \quad {\rm ch}(\pi_{\boldsymbol{\mu}}) = \tilde{P}_{\boldsymbol{\mu}}$.

Theorem. The characteristic map ch is an isometric isomorphism of graded rings.

If we continue with the analogy of the relationship between the symmetric group and symmetric functions, then the characteristic of the irreducible characters of $G_n$ should be some kind of “Schur functions.” Unfortunately their definition will require significantly more notation. So I’ll skip that and just say that we can define functions $S_{\boldsymbol{\lambda}}$. One catch, though, is that the indexing set we use for these Schur functions is not the same as the indexing set for conjugacy classes. The indices ${\boldsymbol{\lambda}}$ can be thought of as partition-valued functions, but on a different domain. But this is not such a big deal.

Theorem. The $S_{\boldsymbol{\lambda}}$ form an orthonormal basis for B. Furthermore, their inverses under the characteristic map are the irreducible characters $\chi^{\boldsymbol{\lambda}}$ of the groups $G_n$. Consequently, the values of the characters are given by the change of bases $S_{\boldsymbol{\lambda}} = \sum_{\boldsymbol{\mu}} \chi^{\boldsymbol{\lambda}}(\boldsymbol{\mu}) \tilde{P}_{\boldsymbol{\mu}}$.

At any rate, I think it is nice that the same kind of setup works for the finite general linear groups as does for the symmetric group, which maybe further justifies the statement that the finite general linear groups are q-analogues of the symmetric groups.

But since these symmetric functions are so horribly complicated, one doesn’t expect to have a nice combinatorial rule for changing from the S basis to the $\tilde{P}$ basis (such as the Murnaghan–Nakayama rule for writing the Schur polynomials in terms of power sum symmetric functions in the symmetric group case). There are some nice cases though. When the conjugacy class $\boldsymbol{\mu}$ corresponds to a unipotent conjugacy class, we can evaluate induced characters from maximal tori T of $G_n$ at $\boldsymbol{\mu}$ to get Green polynomials (up to a sign). And sometimes these induced characters are irreducible (precisely when the stabilizer of the character in the Weyl group of T is trivial).

Green polynomials are more manageable to think about: they arise as the change of basis coefficients when writing power sum symmetric functions as Hall–Littlewood functions (now working just in the ring $\Lambda[t]$ from last time).

That’s basically all I want to say about the connection between symmetric functions and finite general linear groups. There is a more powerful approach to characters of these groups using $\ell$-adic cohomology due to Deligne and Lusztig, and it works more generally for any finite group of Lie type.

-Steven