Home » Mathematics » Algebraic combinatorics » Everything you wanted to know about symmetric polynomials, part V

Everything you wanted to know about symmetric polynomials, part V

This will be the last post on symmetric polynomials, at least for now. (They’ll continue to come up when I get to representation theory and my true love, algebraic geometry, but only as part of other theories.)

I want to discuss the Hall inner product on the symmetric function ring \Lambda and its interaction with the \omega-involution. As a side benefit, we’ll get the “dual” Jacobi-Trudi and Pieri rules, with e and h swapped.

The Inner Product on \Lambda

Let’s define a bilinear form on \Lambda by setting the homogeneous symmetric polynomials to be a dual basis to the monomial symmetric polynomials.

In other words, we define \langle\ ,\ \rangle by setting

\langle m_\lambda, h_\mu \rangle = \delta_{\lambda,\mu}.

By extension, \langle f, h_\mu \rangle is the coefficient of m_\mu in f. On the other hand, \langle m_\lambda, f \rangle is the coefficient of h_\lambda in f. This is not obviously “nice” in any way, at least a priori, but is a convenient way to talk about certain change-of-basis coefficients.

Still, harkening back to my third post on symmetric polynomials, we already know a couple of values of the pairing:

  • \langle e_\lambda, h_\mu \rangle = A_{\lambda \mu}, the number of 0-1 matrices with row sums \lambda and column sums \mu.
  • \langle s_\lambda, h_\mu \rangle = K_{\lambda \mu}, the Kostka number (the number of SSYTs of shape \lambda and content \mu).

By similar reasoning to the matrix-style argument I used for expanding the e‘s in the m-basis, we can find:

  • \langle p_\lambda, h_\mu \rangle = C_{\lambda \mu}, the number of nonnegative integer matrices with i-th row containing one \lambda_i and all other entries zero; and with column sums \mu.
  • \langle h_\lambda, h_\mu \rangle = B_{\lambda \mu}, the number of nonnegative integer matrices with row sums \lambda and column sums \mu.

Oops: something surprising just happened. These last numbers are symmetric in \lambda, \mu, even though we didn’t define our inner product to be symmetric! But, it must be true for the h‘s, and therefore (since they’re a basis) everything else as well:

Proposition. The bilinear form \langle\ ,\ \rangle is symmetric.

But more is true:

Proposition. The Schur polynomials form an orthonormal basis for \Lambda.

Proof. Let’s consider an inner product \langle s_\lambda, s_\mu \rangle. By definition, to compute this we should expand one term in the h basis and the other in the m basis:

\displaystyle{ \langle s_\lambda, s_\mu \rangle = \langle \sum_\alpha L_{\lambda \alpha} h_\alpha, \sum_\beta K_{\mu \beta} m_\beta \rangle = \sum_{\alpha,\beta} L_{\lambda \alpha} K_{\mu \beta} \langle h_\alpha, m_\beta \rangle. }

These last inner products vanish by definition unless \alpha = \beta, so we’re left with

\displaystyle{ \langle s_\lambda, s_\mu \rangle = \sum_\alpha L_{\lambda \alpha} K_{\mu \alpha} = (L \cdot K^T)_{\lambda \mu}, }

which we wish to show equals \delta_{\lambda \mu}. Here, we’re just defining L to be the (hitherto unknown) matrix for converting Schur polynomials to the h basis, and K is the matrix of Kostka numbers, for changing the Schurs to the monomial basis. In other words, we want L and K^T to be inverse matrices.

Equivalently, we could show that the inverse of L, the matrix converting the h basis to the Schur basis, is given by the (transposed) Kostka numbers:

\displaystyle{ h_\lambda = \sum_\mu K_{\mu \lambda} s_\mu.}

In fact, we’ve already shown this (I hinted at it in the second post): this equality is none other than the the RSK correspondence. More specifically, the RSK correspondence is the above equality when written in the m basis. To see this, let’s convert both sides to monomials:

\displaystyle{ \sum_\nu B_{\lambda \nu} m_{\nu} = \sum_{\mu,\nu} K_{\mu \lambda} K_{\mu \nu}m_\nu.}

The coefficient of m_\nu on the one left is the number of nonnegative integer matrices with row sum \lambda and column sum \nu; the coefficient on the right is the number of pairs of SSYTs of the same shape (\mu, summing over all choices), the first tableau having weight \lambda and the second having weight \nu. The fact that these agree is exactly what we proved in the RSK correspondence.

Involving the Involution

Hilarious, I know. Let’s take a look at the effect of the \omega involution on the inner product. We previously established that \omega swaps the e and the h basis. Since we determined above that our inner product is symmetric, we get for free,

Proposition. The \omega involution is an isometry: \langle f,g \rangle = \langle \omega(f), \omega(g) \rangle.

Proof. We can check this on the e,h bases. We know from above that A_{\lambda \mu} = \langle h_\lambda, e_\mu \rangle. On the other hand, \langle \omega(h_\lambda), \omega(e_\mu) \rangle = \langle e_\lambda, h_\mu \rangle = A_{\mu \lambda}. It’s clear from the definition above that $A_{\mu \lambda} = A_{\lambda \mu}$.

Next, let’s see what \omega does to the Schur polynomials, bearing in mind that the result will again be an orthonormal basis. Since we just worked hard establishing the h \to s conversion (with the Kostka numbers), let’s use that. We have

\displaystyle{ h_\lambda = \sum_\nu K_{\nu \lambda} s_{\nu}.}

Applying \omega, we get

\displaystyle{ e_\lambda = \sum_\nu K_{\nu \lambda} \omega(s_{\nu}).}

But last post‘s Pieri rule already told us how to express the e‘s in the Schur basis: we had

\displaystyle{ e_\lambda = \sum_\nu K_{\nu^T \lambda} s_{\nu}.}

Since the matrices (K_{\nu,\lambda}), (K_{\nu^T,\lambda}) are invertible, these equations must match up term-by-term. The matrices are permuted on the Schur polynomial side; if we reindex the second sum, swapping \nu and \nu^T, we get

\displaystyle{ e_\lambda = \sum_\nu K_{\nu \lambda} s_{\nu^T}.}

Now, by inverting the system, we have shown:

Proposition. The \omega involution sends s_\lambda \mapsto s_{\lambda^T}.

In particular, we can now translate the Jacobi-Trudi and Pieri rules from last post to their ‘dual’ forms:

Theorem (Dual Jacobi-Trudi). The Schur polynomial satisfies

\displaystyle{ s_{\lambda^T} = \det( e_{j-1 + \lambda_{n+1-j}}).}

Theorem (Dual Pieri). The Schur polynomials satisfy

\displaystyle{ s_k \cdot s_\lambda = \sum_\mu s_\mu,}

where the sum runs over all partitions \mu \supset \lambda such that \mu/\lambda consists of k boxes in a (collection of) horizontal strip(s). That is, none of the added boxes should be in the same column. (To visualize this, keep in mind that the partition (k) is, itself, a horizontal strip.)

That’s very nice.

The Neglected p Basis

Since this is my last post on symmetric functions, I should briefly mention a few last details on the power symmetric polynomials.

First, by replaying the p-to-e change of basis (the one with the generating functions) with a few extra minus signs, we can get the p-to-h change of basis. It’s basically the same, up to a sign – in particular, it looks almost the same under the \omega involution. By inspection, we can therefore find

\displaystyle{ \omega(p_k) = (-1)^{k-1} p_k.}

This also tells us \omega(p_\lambda) = \pm p_\lambda, since \omega is a ring map.

Second, the inner product satisfies

\displaystyle{ \langle p_\lambda, p_\mu \rangle = z(\lambda)\delta_{\lambda,\mu}.}

In particular, the power symmetric polynomials are self-dual and orthogonal to each other, but are not normal. The coefficient is z(\lambda) = (r_1! \cdots r_n!) \cdot (\lambda_1 \cdots \lambda_k), where r_i is the number of times i occurs in \lambda.

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