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Elementary Jucys-Murphy class sums through S_4

Statement

In $S_{4}$ with $X_{2} = (1 2)$ , $X_{3} = (1 3) + (2 3)$ , $X_{4} = (1 4) + (2 4) + (3 4)$ : the element $e_{1} (X_{2}, X_{3}, X_{4}) = X_{2} + X_{3} + X_{4}$ is the sum of the six transpositions, the permutations of $S_{4}$ with $3$ cycles; the element $e_{2} (X_{2}, X_{3}, X_{4})$ is the sum of the $11$ permutations with $2$ cycles (the three double transpositions and the eight $3$ -cycles); and $e_{3} (X_{2}, X_{3}, X_{4})$ is the sum of the six $4$ -cycles, the permutations with one cycle. The evaluations are central: $e_{1}$ and $e_{3}$ are single class sums, while $e_{2}$ is the sum of the class sums of types $(2, 2)$ and $(3, 1)$ .

Facts & Assumptions

Given: The symmetric group $S_{4}$ acting on ${1, 2, 3, 4}$ and the elements $X_{2} = (1 2)$ , $X_{3} = (1 3) + (2 3)$ , $X_{4} = (1 4) + (2 4) + (3 4)$ of $Z [S_{4}]$ (The Jucys-Murphy elements of the symmetric group algebra).

[F1]

The elementary symmetric polynomials are $e_{1} = x_{1} + x_{2} + x_{3}$ , $e_{2} = x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3}$ , $e_{3} = x_{1} x_{2} x_{3}$ in three variables (The elementary symmetric polynomials $e_{0}, e_{1}, \dots, e_{n}$ ).

[F2]

For $0 \leq s \leq n$ one has $e_{s} (X_{2}, \dots, X_{n}) = \sum_{ρ ⊢ n, ℓ (ρ) = n - s} C_{ρ}^{(n)}$ , the sum of all permutations of $S_{n}$ with exactly $n - s$ cycles (Elementary symmetric Jucys-Murphy evaluations are cycle-count class sums).

[F3]

Cycle type, support and cycle decomposition are as defined in Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type: a permutation of $S_{4}$ has $3$ cycles exactly when it is a transposition, $1$ cycle exactly when it is a $4$ -cycle, and the elements with $2$ cycles are the three double transpositions together with the eight $3$ -cycles.

Proof

technique · direct

1.1F1F3given

The displayed elements are $X_{2} = (1 2)$ , $X_{3} = (1 3) + (2 3)$ and $X_{4} = (1 4) + (2 4) + (3 4)$ ; the products below are computed in the group algebra $Z [S_{4}]$ , in which distinct permutations form a $Z$ -basis.

2.1step 1.1algebra

$X_{2} X_{3} = (1 2) ((1 3) + (2 3)) = (1 2) (1 3) + (1 2) (2 3) = (1 3 2) + (1 2 3)$ .

2.2step 1.1algebra

$X_{2} X_{4} = (1 2) ((1 4) + (2 4) + (3 4)) = (1 2) (1 4) + (1 2) (2 4) + (1 2) (3 4) = (1 4 2) + (1 2 4) + (1 2) (3 4)$ .

2.3step 1.1algebra

$X_{3} X_{4} = ((1 3) + (2 3)) ((1 4) + (2 4) + (3 4)) = (1 3 4) + (1 4 3) + (2 3 4) + (2 4 3) + (1 3) (2 4) + (1 4) (2 3)$ .

2.4F1F3step 1.1algebra

$e_{1} (X_{2}, X_{3}, X_{4}) = X_{2} + X_{3} + X_{4} = (1 2) + (1 3) + (2 3) + (1 4) + (2 4) + (3 4)$ , the six transpositions, i.e. the six permutations with $3$ cycles by [F3].

3.1F1F3step 2.1step 2.2step 2.3algebra

Adding the three products of steps 2.1-2.3 gives $e_{2} (X_{2}, X_{3}, X_{4}) = X_{2} X_{3} + X_{2} X_{4} + X_{3} X_{4} = (1 3 2) + (1 2 3) + (1 4 2) + (1 2 4) + (1 3 4) + (1 4 3) + (2 3 4) + (2 4 3) + (1 2) (3 4) + (1 3) (2 4) + (1 4) (2 3)$ : eight $3$ -cycles and the three double transpositions, i.e. $11$ permutations, all with $2$ cycles by [F3].

3.2F1F3step 1.1step 2.3algebra

$e_{3} (X_{2}, X_{3}, X_{4}) = X_{2} X_{3} X_{4} = (1 2 3 4) + (1 2 4 3) + (1 3 2 4) + (1 3 4 2) + (1 4 2 3) + (1 4 3 2)$ , the six $4$ -cycles, i.e. the permutations with one cycle by [F3]; indeed the product is $X_{2} (X_{3} X_{4})$ and the six products of the three transpositions of $X_{4}$ with $(1 2)$ and the two transpositions of $X_{3}$ are exactly the six listed $4$ -cycles, with no repetitions among them.

4.1F2step 3.1step 2.4step 3.2algebra

By [F2] with $n = 4$ the evaluations $e_{s} (X_{2}, X_{3}, X_{4})$ equal the class sums $\sum_{ℓ (ρ) = 4 - s} C_{ρ}^{(4)}$ for $s = 1, 2, 3$ , which are exactly the three displayed finite sums; each class sum is central because conjugation permutes each conjugacy class. In particular no term cancels and every coefficient is $1$ , as the explicit expansions of steps 3.1 and 3.2 show.

5.1F2step 4.1∎

The case $s = 0$ gives $e_{0} = 1$ and the case $s = 4$ gives $e_{4} = 0$ ; the example exhibits the three nontrivial evaluations and confirms the identity of [F2] at the first size with three nonzero positive-degree elementary evaluations.

Remarks

The counts. $S_{4}$ has $6$ transpositions, $3$ double transpositions, $8$ three-cycles and $6$ four-cycles, so the three sums have $6$ , $3 + 8 = 11$ and $6$ terms; these are the class sizes of the cycle types $2 1^{2}$ , $2^{2}$ , $3 1$ and $4$ .
Centrality. By [F2] each evaluation is a sum of conjugacy-class sums, hence lies in $Z (C [S_{4}])$ . There are four nonidentity conjugacy classes; $e_{2}$ combines two of them, so these three evaluations do not individually list all four class sums.

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Sources

Garsia, Young Seminormal Representation, Murphy Elements and Content Evaluations, UCSD lecture notes (2003), section 3 and Theorem 5.10, printed pp. 18-25 and 51-52 (standard reference, not scraped)
Okounkov-Vershik, A New Approach to the Representation Theory of the Symmetric Groups, Selecta Math. (N.S.) 2 (1996) 581-605; complete arXiv repost math/0503040, section 3, printed pp. 12-16 (standard reference, not scraped)