Products of non-distinct Young–Jucys–Murphy elements

Determine an explicit formula for products of non-distinct Young–Jucys–Murphy elements \(\mathbf{m}_{i_1}\mathbf{m}_{i_2}\cdots\mathbf{m}_{i_k}\), including a formula for powers \(\mathbf{m}_i^k\), while allowing the same permutation in the symmetric-group-algebra expansion to occur with multiplicity.

Background

The paper proves an explicit expansion for products of distinct Young–Jucys–Murphy elements: when $i_1,\ldots,i_k$ are distinct, the product $\mathbf{m}_{i_1}\cdots\mathbf{m}_{i_k}$ equals the sum of permutations whose nonstarter set is exactly $\{i_1,\ldots,i_k\}$. This result also yields formulas for elementary symmetric polynomials in the Young–Jucys–Murphy elements and for sums over permutations of prescribed reflection length.

The unresolved question asks whether an analogous description exists when indices repeat. Repetition changes the combinatorics because expanding a repeated factor can produce the same permutation more than once; the paper illustrates this with $$\mathbf{m}_3^2=2\,\operatorname{id}+\operatorname{cyc}_{1,2,3}+\operatorname{cyc}_{1,3,2}$$.