Posted by: Steven Sam | September 28, 2009

GLFq II: Hall–Littlewood functions

Last time, I discussed how to parametrize the conjugacy classes of the general linear group G over a finite field. In a sense, this group is a q-analogue of the symmetric group, so we might try to imitate the constructions for the symmetric group to get information about G. There’s a nice construction that Frobenius worked out which connects the characters of the symmetric group with the combinatorics of the Schur functions. I’ll briefly summarize the statement. The conjugacy classes of the symmetric group on n letters are parametrized by partitions of n. So we can also parametrize the irreducible characters by partitions as well (though it is not clear how to do this in a “canonical” way a priori). Ignoring the indexing issue (which can be dealt with) and letting \chi^\lambda(\mu) be the irreducible character indexed by \lambda evaluated at the conjugacy class consisting of permutations whose cycle lengths are given by the parts of \mu, then one has s_\lambda(x) = \sum_\mu z_\mu^{-1} \chi^\lambda(\mu) p_\mu(x) where s_\lambda(x) is a Schur function, p_\mu(x) is a power sum (Newton) symmetric function, and z_\mu = 1^{m_1} m_1! 2^{m_2} m_2! \cdots where m_i is the number of times that i appears as a part of \mu (the meaning of z_\mu is that n! z_\mu^{-1} is the size of the conjugacy class index by \mu.)

So the question to ask might be “can we find a similar interpretation for the characters of G?” The answer is yes, but becomes a bit more involved.

Instead of Schur functions, one needs to look at another class of symmetric functions called the Hall–Littlewood functions, but we’ll actually need a much larger ring than the ring of symmetric functions. While the Schur functions are symmetric functions in a set of variables x_1, x_2, \dots usually defined over rational coefficients, the Hall–Littlewood (HL) functions are symmetric functions defined over the ring {\bf Z}[t] (so t is an additional variable which does not affect the definition of “symmetric”.) Let \Lambda[t] denote the symmetric functions in variables x_1, x_2, \dots with coefficients in {\bf Z}[t]. Like the Schur functions, the HL functions are indexed by partitions. The definition of the Hall–Littlewood function indexed by \lambda in n variables x_1, \dots, x_n is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = \sum_{w \in S_n / S_n^\lambda} w\left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{\lambda_i > \lambda_j} \frac{x_i - tx_j}{x_i - x_j} \right) .

Here S_n is the symmetric group on n letters, and S_n^\lambda \subseteq S_n is the subgroup of permutations w such that \lambda_{w(i)} = \lambda_i for all i. From this, one can see that P_\lambda(x_1, \dots, x_n; 1) = m_\lambda(x_1, \dots, x_n) where m_\lambda denotes the monomial symmetric function which is the sum of all of the distinct terms x_{w(1)}^{\lambda_1} \cdots x_{w(n)}^{\lambda_n} as w ranges over all permutations of n.

Set [\lambda]_t! = [\lambda_1]_t! \cdots [\lambda_n]_t!. An equivalent definition for P_\lambda is

\displaystyle P_\lambda(x_1, \dots, x_n; t) = [\lambda]_t!^{-1} \sum_{w \in S_n} w \left( x_1^{\lambda_1} \cdots x_n^{\lambda_n} \prod_{i < j} \frac{x_i - tx_j}{x_i - x_j} \right),

from which one can deduce that P_\lambda(x_1, \dots, x_n; 0) = s_\lambda(x_1, \dots, x_n) from the Weyl character formula.

I just want to state some of the properties that we will need later without giving too many details. For proofs, one can consult Macdonald's book Symmetric Functions and Hall Polynomials. From the first definition, one can deduce that these functions enjoy a stability property:

P_\lambda(x_1, \dots, x_n, 0; t) = P_\lambda(x_1, \dots, x_n; t),

and hence one can define P_\lambda(x;t) in infinitely many variables x_1, x_2, \dots by taking an inverse limit. Since the P_\lambda are symmetric functions, we can write

P_\lambda(x;t) = \sum_\mu w_{\lambda, \mu} (t) s_\mu(x)

for some polynomials w_{\lambda,\mu}(t). In fact, w_{\lambda, \lambda}(t) = 1 and w_{\lambda,\mu}(t) = 0 unless \lambda \ge \mu (dominance order), so the change of basis matrix from P_\lambda to s_\lambda is upper unitriangular, which implies that the P_\lambda(x;t) form a {\bf Z}[t]-basis of \Lambda[t].

The inverse of this change of basis is very interesting. In this case, write s_\lambda(x) = \sum_\mu K_{\lambda, \mu}(t) P_\mu(x;t). The K_{\lambda, \mu}(t) are the Kostka–Foulkes polynomials. Since P_\mu(x;1) = m_\mu(x), we see that K_{\lambda, \mu}(1) = K_{\lambda, \mu} are the Kostka numbers. It is a fact that the Kostka–Foulkes polynomials are in fact polynomials, and they have nonnegative integers. I hope to write a post about these at some point.

We will also need augmentations of these functions in the next post. First set

\displaystyle b_\lambda(t) = \prod_{i \ge 1} [(1-t)(1-t^2) \cdots (1-t^{m_i(\lambda)})]

where m_i(\lambda) is the multiplicity with which i appears in \lambda. Then define

Q_\lambda(x; t) = b_\lambda(t) P_\lambda(x; t).

Although we won’t use them, let me mention skew Hall–Littlewood functions. Define an inner product on \Lambda[t] by declaring that \langle P_\lambda(x;t), Q_\mu(x;t) \rangle = \delta_{\lambda, \mu}. Then we can define skew Hall–Littlewood functions for partitions \mu \subseteq \lambda via

\langle Q_{\lambda/\mu}(x;t), P_\nu(x;t) \rangle = \langle Q_\lambda(x;t), P_\mu(x;t) P_\nu(x;t) \rangle

\langle P_{\lambda/\mu}(x;t), Q_\nu(x;t) \rangle = \langle P_\lambda(x;t), Q_\mu(x;t) Q_\nu(x;t) \rangle.

From this definition, setting t=0 gives back the skew Schur functions s_{\lambda/\mu}(x) (since they are defined in a similar way). The weird thing, however, is that the skew Schur functions only depend on the shape \lambda/\mu, whereas the skew Hall–Littlewood functions remember both \lambda and \mu. One can write down a rather explicit formula for Q_{\lambda/\mu} in terms of semistandard tableaux which shows that the function depends on both \lambda and \mu (but this is only seen in the powers of t, and not the x_i), but I will omit this so that I can wrap this post up.

Let me just end with some other specializations of t that are important. When \lambda is a strict partition (i.e., the nonzero parts are distinct) then setting t=-1 gives P_\lambda(x;-1) = 2^{\ell(\lambda)} Q_\lambda where \ell(\lambda) is the number of parts, and Q_\lambda are the Schur Q-functions, which are important for the projective representation theory of the symmetric group (maybe a future topic). Also, specializations at t=q^{-1} for q a prime power are related to Hall algebras, which are used to keep track of extensions between finite Abelian groups.

In the next post, I’ll discuss the connection between characters of {\bf GL}_n({\bf F}_q) and symmetric functions.


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