Polynomials and Representations XIII

Skew Diagrams

If we multiply two elementary symmetric polynomials e_\lambda and e_\mu, the result is just e_\nu, where \nu is the concatenation of \lambda and \mu sorted. Same holds for h_\lambda h_\mu. However, we cannot express s_\lambda s_\mu in terms of s_\nu easily, which is unfortunate since the Schur functions are the “preferred” basis, being orthonormal. Hence, we define the following.

Definition. A skew Young diagram is a diagram of the form \lambda/ \mu, where \lambda and \mu are partitions and \lambda_i \ge \mu_i for each i.

E.g. if \lambda = (7, 5, 3) and \mu = (5, 2, 1), then


Note that the same skew Young diagram can also be represented by \lambda'/\mu' where \lambda' = (8, 6, 4, 1) and \mu' = (6, 3, 2, 1). These two Young diagrams are considered identical.

Definition. A skew semistandard Young tableau (skew SSYT) is a labelling of the skew Young diagram with positive integers such that each row is weakly increasing and each column in strictly increasing. Now \lambda/\mu is called the shape of the tableau and its type is given by \alpha where \alpha_i is the number of times i appears.

E.g. the following is a skew SSYT of the above shape. Its type is (4, 1, 1, 1).



Skew Schur Polynomials

Definition. The skew Schur polynomial corresponding to \lambda/\mu is given by:

\displaystyle s_{\lambda/\mu} := \sum_{\text{shape}(T) = \lambda/\mu} x^T

where x^T is \prod_i x_i^{\text{\# of } i \text{ in } T}. E.g. the above diagram gives x^T = x_1^4 x_2 x_3 x_4.

The proof for the following is identical to the case of Schur polynomials.

Lemma. The skew Schur polynomial s_{\lambda/\mu} is symmetric.

Indeed, one checks easily that the Knuth-Bendix involution works just as well for skew Young tableaux.


So the number of skew SSYT of shape \lambda/\mu and type \nu is unchanged when we swap \nu_i and \nu_{i+1}.


For \lambda = (3, 2) and \mu = (1), we have:


The following result explains our interest in studying skew Schur polynomials.

Lemma. The product of two skew Schur polynomials is a skew Schur polynoial.

For example, we have:


It remains to express s_{\lambda/\mu} as a linear combination of s_\nu, where |\nu| = |\lambda| - |\mu|.


Littlewood-Richardson Coefficients

Recall that we have Pieri’s formula h_r s_\lambda = \sum_\nu s_\nu, where \nu is summed across all diagrams obtained by adding r boxes to \lambda, such that no two are on the same column. Repeatedly applying this gives us:

\displaystyle h_\mu s_\lambda = \sum_{\nu_k} \sum_{\nu_{k-1}} \ldots \sum_{\nu_1} s_{\nu_k}

where \nu_0 := \lambda and \nu_{i+1} is obtained from \nu_i by adding \mu_i boxes such that no two lie in the same column. Hence, the number of occurrences for a given skew Young diagram \nu' is the number of skew SSYT’s with shape \nu' and type \mu.


If \mu = (4, 2, 1) and \lambda = (5, 4, 3), here is one way of appending 4, 2, 1 boxes in succession:


which corresponds to the following skew SSYT:


This gives us the tool to prove the following.

Theorem. For any f\in \Lambda^{(d)} with d = |\lambda| - |\mu|, we have:

\displaystyle \left< s_{\lambda/\mu}, f\right> = \left< s_\lambda, f s_\mu\right>

so the linear map s_\lambda \mapsto s_{\lambda/\mu} is left adjoint to multiplication by s_\mu.


It suffices to prove this for all f = h_\nu where \nu\vdash d. Since \{h_\lambda\} is the basis dual to \{m_\lambda\}, the LHS is the coefficient of m_\nu in expressing s_{\lambda/\mu} in terms of monomial symmetric polynomials. By definition of s_{\lambda/\mu}, this is equal to the number of skew SSYTs with shape \lambda/\mu and type \nu; we will denote this by K_{\lambda/\mu, \nu}, the skew Kostka coefficient.

By the reasoning above, when h_\nu s_\mu is expressed as a linear combination of Schur functions, the coefficient of s_\lambda is also K_{\lambda/\mu, \nu}. Since the Schur functions are orthonormal, we are done. ♦


The theorem is still true even when \lambda_i \ge \mu_i does not all hold, if we take s_{\lambda/\mu} = 0. Indeed, by reasoning with Pieri’s rule again, every Schur polynomial s_\lambda occuring in h_\nu s_\mu must have \lambda_i\ge \mu_i for all i.

Definition. When we express:

\displaystyle s_\mu s_\nu = \sum_{\lambda} c_{\mu\nu}^\lambda s_\lambda,

the values c_{\mu\nu}^\lambda = \left< s_\mu s_\nu, s_\lambda\right> are called the Littlewood-Richardson coefficients.

By the above theorem, this equals \left< s_{\lambda/\mu}, s_\nu\right>. One can calculate this using the Littlewood-Richardson rule, which we will cover later.

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