# $S_n$ action on the sequences of transpositions

It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i < b_{i+1}$ for any $i$. Another way to say it is that the sum $\sum_{j = 1}^n e_j(J_2,\ldots,J_n)$ of elementary symmetric polynomials evaluated on Jucys-Murphy elements is a central element of the group algebra $\mathbb C[S_n]$, that can be presented as $\sum_{\tau \in S_n} \tau$. We refer to a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i < b_{i+1}$ for any $i$ to as a strictly monotone factorization.

Thus, there is a natural action of $S_n$ on the set of strictly monotone factorizations, isomorphic to the action of $S_n$ on itself by conjugations. This action can be completely described in the terms of strictly monotone factorizations only.

We may also consider the set $M_p$ of weakly monotone factorizations, i.e. the set of sequences $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i < b_i$ and $b_i \le b_{i+1}$. This set is naturally related to the complete homogeneous symmetric polynomial $h_p$ evaluated on Jucys-Murphys. The set $M_p$ admits a natural map $\pi\colon M_p \to S_n$ -- the product of the elements of the sequence in the prescribed order.

Is there a natural structure of $S_n$-space on $M_p$, such that $\pi$ is a morphism of $S_n$-spaces, where $S_n$ acts on itself by conjugations?

• Barring that, what if we replace $b_i \leq b_{i+1}$ by $b_i \geq b_{i+1}$ ? – darij grinberg Dec 23 '16 at 7:30
• When you say "This action can be completely described in the terms of stictly monotone factorizations only", do you have a specific description in mind? I see how to describe the action of a simple transposition $s_k$: Namely, it replaces all $k$'s in the transpositions by $k+1$'s and vice versa. Then, it checks if the result is still a monotone factorization. There are three possible cases in which it isn't, and they can be dealt with separately (by "local straightening rules"). But this doesn't sound like a very useful description :( – darij grinberg Dec 23 '16 at 7:47
• @darijgrinberg For your first comment -- the change of monotonicity direction really changes nothing, there is a natural 1-1 correspondence between monotonically increasing and monotonically decreasing sequences, which corresponds to reversing the order of elements in the sequence. – Max Karev Dec 23 '16 at 9:17
• @darijgrinberg For the second one -- conjugate a sequence termwise by a permutation. The monotonicity condition generally fails, but there exists a set of "local straightening rules" admitting a description in terms of the braid group action on the set of sequences of permutations. The question was motivated by Hurwitz numbers computation problem, which can be stated as a count of numbers of sequences of transpositions, without any additional restriction. The $S_n$-action on Hurwitz factorizations is obvious, so it is natural to ask, is there a natural action in the monotone case as well. – Max Karev Dec 23 '16 at 9:25
• @darijgrinberg Returning again to the "local straightening rules". The most convenient description of the action is the following. Conjugate a strictly monotone factorization termwise by a chosen element of $S_n$. The resulting factorization, in general, fails to be strictly monotone. But its orbit with respect to the action of the braid group on the set of factorization contains a unique strictly monotone representative. – Max Karev Dec 24 '16 at 10:14