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Everything you wanted to know about symmetric polynomials, part IV


Alternating and Symmetric Polynomials

Consider the following recipe for building symmetric polynomials, using alternating polynomials. Consider the Vandermonde determinant

\displaystyle{  \Delta(x_1, \ldots, x_n) = \left|\begin{matrix}  1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\  1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\  & \vdots & & \vdots & \\  1 & x_n & x_n^2 & \cdots & x_n^{n-1}  \end{matrix}\right| = \prod_{i < j} (x_j - x_i).  }

To see the last equality, note that the determinant is zero if x_i = x_j for any i \ne j, so it is divisible by (x_j-x_i), and all these factors are distinct. By degree-counting (the polynomial is evidently homogeneous of degree n \choose 2), this is the whole thing, up to a scalar. Finally, to get the scalar, we do some computation (e.g. plugging in convenient values like x_i = i).

If, instead of the row (1 \ x_i\ x_i^2\ \cdots\ x_i^{n-1}), we used some other sequence of polynomials, like (f_1(x_i)\ f_2(x_i)\ \cdots \ f_{n-1}(x_i)), the result would still be alternating in the x_i‘s, so it would still be divisible by the product above. However, the degree-counting argument might no longer be relevant. (For example, if we use (x_i\ x_i^2\ \cdots\ x_i^n), then the result is the original Vandermonde determinant times x_1 \cdots x_n.)

Still, we can divide out the Vandermonde determinant, and (surprise!) the result will be a symmetric polynomial, since the sign-change is “divided out” as well.

For example, we might as well consider


for any sequence of exponents c_1, \ldots, c_n. Of course, the determinant will only be nonzero if they’re all distinct, and by rearranging we can assume c_1 < c_2 < \cdots < c_n. It's worth noting that the "Vandermonde case" of c_j = j-1 is the simplest one – in fact, it is the unique(!) alternating polynomial of minimal degree, up to scaling. We may as well describe the others in terms of it: let's define

\Delta(\lambda) = \det(x_i^{j-1+\lambda_{n+1-j}}),

for any partition \lambda = \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0. (So the “zero partition” corresponds to the Vandermonde determinant itself.) Now we get a class of symmetric polynomials

\displaystyle{ s_\lambda = \frac{\Delta(\lambda)}{\Delta(x_1, \ldots, x_n)} },

each of which is homogeneous of degree |\lambda|. We have the following theorem:

Theorem. The ratio of alternants s_\lambda is the Schur polynomial.

In fact, this was the original definition of the Schur polynomials, before Young tableaux were used. The proof is by clearing the \Delta(\lambda) denominator and comparing monomial coefficients. As usual, it boils down to one particular computation, namely:

Lemma. Let \lambda be a partition, \alpha \in \mathbb{Z}^n, and \rho = (0,1,\ldots, n-1). Then

\displaystyle{ \sum_{\sigma \in S_n} (-1)^\sigma K_{\lambda, \sigma \cdot (\alpha+\rho) - \rho}  = \begin{cases} 1 & \lambda = \alpha \\ 0 & \text{ otherwise.} \end{cases}}

This is proved using a sign-canceling involution on SSYT(\lambda) - \{T_*\}, where T_* is the “highest-weight tableau”, whose i-th row is all is. It’s in the same spirit as the Bender-Knuth involution: it changes the weight of each tableau from w(T) to s_i(w(T) + \rho) - \rho, where s_i is a transposition (i is the first row where T differs from T_*.) This makes all the terms cancel, except possibly the T_* term itself (if it appears at all — it has weight \lambda). Finally, we check that T_* can only show up in the \sigma = id term, and then only if \lambda = \alpha.

Jacobi-Trudi and Pieri

It’s time to discuss a couple of big formulas involving the Schur polynomials. The first, the Jacobi-Trudi formula, is both theoretically-useful, as it expresses s_\lambda as a polynomial in the h_k‘s, and has a beautiful proof from graph theory. The second, called the Pieri rule, is a useful formula for multiplying complicated Schur polynomials by simple ones; in particular, it’s way easier to use than the Littlewood-Richardson rule.

Theorem (Jacobi-Trudi). The Schur polynomial satisfies

s_\lambda = \det (h_{\lambda_i + j - i}),

that is, it is the determinant of the matrix whose i-th row is

(\cdots h_{\lambda_i + t} \ h_{\lambda_i + t+1} \cdots),

with h_{\lambda_i} down the main diagonal.

The proof of Jacobi-Trudi uses the Lindstrom-Gessel-Viennot Lemma, which expresses the determinant as a weighted sum over collections of certain nonintersecting paths through a graph. Whenever the paths intersect, we flip them around each other and they cancel out. It’s a lovely graph-theoretic argument. I won’t attempt to reproduce the proof here (since it really needs a picture), but it’s worth knowing, particularly as the same approach gives rise to other determinantal formulas as well — generalizations of Jacobi-Trudi, for example, and the Cauchy-Binet formula (which directly establishes the multiplicativity / functoriality of the determinant).

This identity gives another, circuitous proof of the fact that the homogeneous symmetric polynomials h_k generate \Lambda as a ring: we know independently that the Schur polynomials are an additive basis, and now the Jacobi-Trudi identity expresses the Schurs as polynomials in the h_k‘s. There will be some other useful consequences later on. On a side note, since h_k = s_k, we can also think of this fact as showing that the s_k‘s generate \Lambda as a ring.

One final fact: although the Littlewood-Richardson rule (briefly discussed last post) gives a general algorithm for multiplying Schur polynomials, there are other rules for special cases. One of these is the Pieri rule, which is much easier to understand, and computes s_{1,\ldots,1} \cdot s_\lambda (= e_k \cdot s_\lambda). I’ll discuss it in detail, as it gives some of the ‘flavor’ for working with alternating polynomials.

Pieri Rule. We have

\displaystyle{s_{1^k} \cdot s_\lambda = \sum_{\nu} s_\nu,}

where the sum runs over all \nu \supset \lambda such that \nu/\lambda consists of k boxes in a (collection of) vertical strip(s). That is, no two of the boxes we added to \lambda to get \nu should be in the same row. (To remember this, keep in mind that the partition (1^k) is, itself, a vertical strip).

Proof. We’ll use the ratio of alternants formula somewhat indirectly: we consider s_{1^k} \cdot s_\lambda \cdot \Delta(x_1, \ldots, x_n) and decompose the result into a sum of terms of the form \Delta(\nu).

First, let’s determine the coefficients of monomials in s_{1^k} \cdot s_\lambda \cdot \Delta(x_1, \ldots, x_n) = s_{1^k} \cdot \Delta(\lambda). This is an alternating polynomial, so the coefficient of x_1^{a_1} \cdots x_n^{a_n} vanishes if two a_i‘s are equal. So, up to sign changes, we can assume a_1 < \cdots < a_n.

Now, we know that \Delta(\lambda) is the determinant with entries x_i^{j-1+\lambda_{n+1-j}}. And s_{1^k} = e_k is the sum of all squarefree monomials of total degree k. If we expand the determinant and the product, and consider the monomials we get, we see that we’re trying to get exponents

(a_1, \ldots, a_n) = \sigma \cdot (\rho + \lambda^R) + (0,1,\ldots,1,0),

where \rho = (0,1,\ldots,n-1), \lambda^R means the reversal of \lambda (since we’re counting upwards), \sigma is a permutation and the third summand has k ones and all other entries 0.

The key observation is: \rho + \lambda is in strictly-increasing order, and adding k ones here and there can, at most, weaken certain inequalities. It can never reverse them. In particular, if the end result is (a_1, \ldots, a_n), which is again in strictly-increasing order, then:

  • the permutation \sigma must be the identity,
  • the k ones must only be in entries j so that a_j < a_{j+1}.

In other words, either a_j = \rho_j + \lambda_{n+1-j} or \rho_j + \lambda_{n+1-j} + 1, and the latter case happens precisely when "there's room for it" without breaking the inequality.

Thinking pictorially, this means a corresponds to a new partition, that is,

a = \rho + \nu^R,

with \nu obtained by adding k new boxes to \lambda in a “vertical strip” (since each row increments by at most 1), and the placement rule is just guaranteeing that the new boxes result in a valid Young diagram. Moreover, \nu only arises once this way, since it comes from a unique choice of the k ones.

To complete the proof, we double-check that this is equal to \sum_\nu \Delta(\nu), the sum over the relevant partitions \nu. This is more or less immediate (it’s alternating, restrict to monomials with strictly increasing exponents, compare coefficients).

Corollary. The change-of-basis from e to the Schur polynomials is given by

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

Proof. By definition, e_\lambda = e_{\lambda_1} \cdots e_{\lambda_k}. Since e_i = s_{1^i}, we can apply the Pieri rule repeatedly. The coefficient of s_\mu is, then, the number of ways of getting \mu by a putting together a sequence of k vertical strips of the appropriate sizes, starting with an empty tableau.

This is the same data needed to construct a “row-strict” tableau of shape \mu and weight \lambda (imagine filling in the i-th vertical strip with all i‘s). In other words, (by transposing, so as to be column-strict) the coefficient is the number of SSYTs of shape \mu^T and weight \lambda.

Some final remarks

By now, we’ve computed many changes of basis: s, e to m, and h,p to e (somewhat indirectly, since I only wrote out a polynomial change of basis, not an additive one). Now we also know e to s.

Ignoring the somewhat-less-important power symmetric polynomials, we have almost every change of basis: we’re just missing h to m and h to s. (It’s also easy to get p to m.) Stay tuned!


1 Comment

  1. […] Everything you wanted to know about symmetric polynomials, part IV […]

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