Posted by: Steven Sam | January 18, 2010

Littlewood-Richardson coefficients for classical groups

One thing I’ve been thinking about lately is the tensor product multiplicities for the classical groups. The case of the general (special) linear group is the Littlewood-Richardson rule, so I wanted to discuss how to use these numbers to gain some information for the orthogonal and symplectic groups.

1. Notation

First, let’s get some notation out of the way. We’ll work over a field K of characteristic 0, and E will denote a vector space equipped with a nondegenerate symplectic or orthogonal form \omega. Write \dim E = 2n + \tau where \tau \in \{0,1\}. We’ll assume that E has a basis \{e_1, \dots, e_{2n+\tau}\} such that \omega(e_i, e_{2n+1+\tau-i}) = \pm \omega(e_{2n+1+\tau-i}, e_i) = 1 for i=1,2,\dots,n+\tau and such that all other pairings are 0. If K is algebraically closed, this always happens. We let G be the subgroup of \mathbf{SL}(E) which preserves this form, i.e., the subgroup of all g with determinant 1 such that \omega(v,w) = \omega(gv, gw) for all v and w. Let’s use the Cartan-Killing classification to name these groups as follows: when \omega is orthogonal and \dim E is odd (resp. even), we say that G is type B (resp. type D), and when \omega is symplectic, we say that G is type C.

We take T to be a subgroup of G consisting of diagonal matrices. This is a maximal torus, and the matrices look like {\rm diag}(x_1, \dots, x_n, x_n^{-1}, \dots, x_1^{-1}) in the case that \tau = 0 and look like {\rm diag}(x_1, \dots, x_n, 1, x_n^{-1}, \dots, x_1^{-1}) otherwise. For uniformity of notation, let’s write these matrices as {\rm diag}(x_1, \dots, x_n, \dots). The character group of T can be identified with \mathbf{Z}^n by associating the character {\rm diag}(x_1, \dots, x_n, \dots) \mapsto x_1^{\epsilon_1} \cdots x_n^{\epsilon_n} with the element (\epsilon_1, \dots, \epsilon_n) \in {\mathbf Z}^n.

2. Representations of classical groups

With the identification above, we would like to know which characters are dominant weights. We can take our Borel subgroup to be the upper triangular matrices inside of G, so that we have picked a subset of positive roots. In type B and C, a weight \lambda is dominant if and only if \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n \ge 0, and in type D we have a slightly relaxed condition \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_{n-1} \ge |\lambda_n|.

We can construct the representations with these highest weights as certain submodules of representations of the general linear group {\bf SL}(E). The Weyl construction goes as follows. First, given any partition \lambda with at most 2n+\tau parts, we can find a copy of the irreducible representation of {\bf SL}(E) of highest weight \lambda inside of E^{\otimes N} where N = |\lambda| = \sum_i \lambda_i.

Call this representation {\bf S}_\lambda(E). Given our form \omega, and integers (i,j) such that 1 \le i < j \le N, we have a contraction map E^{\otimes N} \to E^{\otimes (N-2)} defined by v_1 \otimes \cdots \otimes v_N \mapsto \omega(v_i, v_j) v_1 \otimes \cdots \otimes \hat{v_i} \otimes \cdots \otimes \hat{v_j} \otimes \cdots \otimes v_N where the hat means we have removed those two vectors. Define {\bf S}_{[\lambda]}(E) to be the intersection of {\bf S}_\lambda(E) with the kernels of all possible contraction maps. Then we have the following. Let \lambda be a partition with at most n parts. In types B and C, {\bf S}_{[\lambda]}(E) is an irreducible representation of G with highest weight \lambda. The same is true in type D if \lambda_n = 0. Otherwise, in type D with \lambda_n > 0, we have that {\bf S}_{[\lambda]}(E) is the direct sum of two irreducible representations of G, one with highest weight \lambda, and the other with highest weight \lambda^- = (\lambda_1, \dots, \lambda_{n-1}, -\lambda_n).

Finally, let me just mention that these three groups are semisimple, so all finite dimensional representations break up into a direct sum of irreducible representations.

3. Tensor products

Let c_{\lambda, \mu}^\nu denote the Littlewood-Richardson coefficients for the general linear group. These can be defined by the tensor product decomposition (as {\bf GL}(V) representations)

\displaystyle {\bf S}_\lambda(V) \otimes {\bf S}_\mu(V) \cong \bigoplus_\nu {\bf S}_\nu(V)^{\oplus c_{\lambda, \mu}^\nu}

when \dim V \ge \ell(\lambda) + \ell(\mu).

Suppose that we want a decomposition of the tensor product

\displaystyle {\bf S}_{[\lambda]}(E) \otimes {\bf S}_{[\mu]}(E) = \sum_{\nu} {\bf S}_{[\nu]}(E)^{\oplus N^{\nu}_{\lambda, \mu}}.

Under the assumption that n \ge \ell(\lambda) + \ell(\mu), the Newell-Littlewood formula gives that

\displaystyle N^\nu_{\lambda, \mu} = \sum_{\alpha, \beta, \gamma} c^\lambda_{\alpha, \beta} c^\mu_{\beta, \gamma} c^\nu_{\alpha, \gamma}.

Note that the formula does not take into account which group we were dealing with. This is surprising (at least to me)! However, the answer is really limiting because we made a largeness assumption on n. For small values of n, the formula above can still be made valid if we reinterpret what {\bf S}_{[\nu]}(E) means when \ell(\nu) > n. I’ll say something about these modification rules in the next section.

4. Modification rules

We want to reinterpret {\bf S}_{[\nu]}(E) in the case that \ell(\nu) > n. In each case, we will either replace it by 0, or \pm {\bf S}_{[\eta]}(E) where \ell(\eta) \le n. In this case, let’s say that \eta is the modification of \nu. Let's describe these modification rules separately for each group.

4.1. Type {\rm B}_n

Let \lambda be a partition and draw its Young diagram. Let \lambda'_i be the number of boxes in the ith column. Set k_i = \lambda'_i if n \ge \lambda'_i - i + 1 and k_i = 2n + 2i - 1 - \lambda'_i otherwise. Here’s a way to picture what the k_i mean. If it exists, put a horizontal dashed line through the middle of the (n + i – 1)st box of the ith column for all i. Then fold the column along that dashed line and delete any parts with overlap. If the part below the dashed line is longer than the part above, the new column becomes negative.

Now set t_i = k_i - i + 1. If the t_i are not distinct, then the modification of \lambda is 0. Otherwise rearrange the t_i in decreasing order using a permutation \sigma, add 1 to the second largest, add 2 to the third largest, etc. These new numbers are the column lengths of a partition \mu, and the modification of \lambda is (-1)^{\sigma} \mu where (-1)^\sigma is the sign of the permutation \sigma.

The C and D case are similar, but we need to define the k_i slightly differently. Both also have interpretations in terms of folding Young diagrams. So we’ll just mention how to define the k_i in these two cases.

4.2. Type {\rm C}_n

Set k_i = \lambda'_i if n \ge \lambda'_i - i + 1 and k_i = 2n + 2i - \lambda'_i otherwise.

4.3. Type {\rm D}_n

Set k_i = \lambda'_i if n \ge \lambda'_i - i + 1 and k_i = 2n + 2i - \lambda'_i - 2 otherwise.

References.

  1. Kazuhiko Koike and Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type B_n, C_n, D_n, J. Algebra 107 (1987), no. 2, 466–511.
  2. D. E. Littlewood, Products and plethysms of characters with orthogonal, symplectic and symmetric groups, Canad. J. Math. 10 1958 17–32.

-Steven

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