Math 202C: Lecture 11

*** Problems in this lecture due 05/03/2026 at 23:59 ***

In Lecture 10, we proved that the Jucys-Murphy specialization \Xi_n \colon \mathcal{S}_n \to \mathcal{Z}_n is a surjection of the algebra \mathcal{S}_n=\mathbb{C}[x_1,\dots,x_n]^{S_n} of symmetric polynomials onto the center \mathcal{Z}_n=\mathcal{C}(S_n) of the convolution algebra of the symmetric group S_n. That is, for every central element |A\rangle \in \mathcal{C}(S_n) there exists a symmetric polynomial f \in \mathcal{S}_n such that |A\rangle = f(\Xi_n), where

f(\Xi_n) = f(|J_1\rangle,\dots,|J_n\rangle)

is the evaluation of f on the Jucys-Murphy elements

|J_t\rangle = \sum\limits_{1 \leq s < t} |st\rangle, \quad 1 \leq t \leq n.

Equivalently, combining the fundamental theorem of symmetric polynomials with the fact that

e_r(\Xi_n) = |L_r\rangle, \quad 0 \leq r \leq n-1,

where

|L_r\rangle = \sum\limits_{\substack{\pi \in S_n \colon |\pi|=r}} |\pi\rangle

is the sum of all elements on level r of the Cayley graph G_n=(S_n,T_n) of S_n as generated by the class T_n = C_1(n) of transpositions, we can say that there exists a (not necessarily symmetric) polynomial g \in \mathcal{P}_n=\mathbb{C}[x_1,\dots,x_n] such that

|A\rangle = g(|L_0\rangle,|L_1\rangle,\dots,|L_{n-1}\rangle).

So the two takeaways are:

  • Every central element is a symmetric polynomial in the Jucys-Murphy elements |J_t\rangle;
  • Every central element is a polynomial in the level elements |L_r\rangle.

Symbolically, we have

\mathcal{Z}_n=\mathbb{C}[|J_1\rangle,\dots,|J_n\rangle]^{S_n}=\mathbb{C}[|L_0\rangle,\dots,|L_{n-1}\rangle],

which means that we have two surjective specializations

\mathcal{S}_n \to \mathcal{Z}_n \quad\text{and}\quad \mathcal{P}_n \to \mathcal{Z}_n,

the first obtained by substituting |J_1\rangle,\dots,|J_n\rangle for the variables of symmetric polynomials in n variables, and the second obtained by substituting |L_0\rangle,\dots,|L_{n-1}\rangle for the variables of arbitrary polynomials in n variables.

This begs the question: what happens when we substitute |J_1\rangle,\dots,|J_n\rangle for the variables of arbitrary polynomials in n variables? More precisely, recall that \mathcal{J}_n is the commutative subalgebra of \mathcal{C}(S_n) generated by

\mathcal{Z}_1 \cup \dots \cup \mathcal{Z}_n,

where \mathcal{Z}_k is the center of the subalgebra \mathcal{C}(S_k) of \mathcal{C}(S_n) spanned by permutations which fix the points k+1,\dots,n. Our original proof of the fact that the JM-elements commute was simply to point out that they lie in the commutative algebra \mathcal{J}_n. We may thus regard the JM-specialization as an algebra homomorphism

\Xi_n \colon \mathcal{P}_n \longrightarrow \mathcal{J}_n

defined by

f(x_1,\dots,x_n) \mapsto f(\Xi_n)=f(|J_1\rangle,\dots,|J_n\rangle).

Since we know that \Xi_n surjects \mathcal{S}_n onto \mathcal{Z}_n, the natural guess is that it surjects \mathcal{P}_n onto \mathcal{J}_n.

Theorem 11.1. The JM-specialization is a surjection of \mathcal{P}_\mathbb{C}[x_1,\dots,x_n] onto \mathcal{J}_n = \mathrm{Alg}(\mathcal{Z}_1 \cup \dots \cup \mathcal{Z}_n).

In fact, this is a direct consequence of what we have already established, which gives that

\Xi_k \colon \mathcal{S}_k \to \mathcal{Z}_k

is surjective for each k.

Problem 11.1. Write out a careful proof of Theorem 11.1.

Note to those who are perusing one of the recommended texts, namely this one: the material we have established so far takes us up to and through the proof of Theorem 4.4.5, as well as the result there called Olshanskis’s theorem, although our arguments have been quite different.

Published by Jonathan Novak

Mathematician at UC San Diego.

Leave a Reply