# Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials

$$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$$Let $$\mathfrak{S}_n$$ be the symmetric group, $$\SYT_n$$ be the set of standard young tableaux of size $$n$$.

For $$u\in \mathfrak{S}_n$$, let $$\RSK:\mathfrak{S}_n\to \SYT_n^2$$ denote the Robinson-Schensted-Knuth correspondance.

Let $$P_{u,w}(q)$$ be the Kazhdan-Lusztig polynomial and $$\mu_{u,w}=[q^{{(l(w)-l(u)-1)}/{2} }]P_{u,w}(q)$$.

Let $$\evac:\SYT_n\to \SYT_n$$ be Schutzenberger's involution.

Question: I am looking for a proof of the following proposition that Schutzenberger's involution preserves the $$\mu$$-coefficient:

Take $$u,w\in \mathfrak{S}_n$$, and let $$(P_1,Q_1)=\RSK(u)$$, $$(P_2,Q_2)=\RSK(w)$$. If we set $$\begin{gather*} u'=\RSK^{-1}(\evac(P_1),\evac(Q_1)) \\ w'=\RSK^{-1}(\evac(P_2),\evac(Q_2)), \end{gather*}$$ then we have $$\mu(u,w)=\mu(u',w').$$

• In the title: "evacuation"? – paul garrett Dec 13 '18 at 22:42
• Perhaps it follows from the fact that evacuation is basically conjugation by $w_0$, which is an automorphism of the Dynkin diagram? – Sam Hopkins Dec 13 '18 at 22:54
• You say "For $u \in \mathfrak S_n$, let $RSK$ …", but your definition of RSK doesn't refer to $n$. – LSpice Dec 14 '18 at 1:23

Let $$P^* = evac(P)$$. As noted above, if $$RSK(u)=(P,Q)$$, then $$RSK(w_0uw_0) = (P^*,Q^*)$$. Conjugation by $$w_0$$ induces an automorphism of the Hecke algebra sending $$T_x \mapsto T_{w_0 x w_0}$$ and $$c_x \mapsto c_{w_0 x w_0}$$, from which the result you want follows. However you might be interested that something stronger is true: if $$u=RSK^{-1}(P,Q)$$ and $$\sigma(u) = RSK^{-1}(P^*,Q)$$ then $$\mu(u,v)=\mu(\sigma(u),\sigma(v))$$. Similarly if $$\rho(u) = RSK^{-1}(P,Q^*)$$ then $$\mu(u,v)=\mu(\rho(u),\rho(v))$$.
This follows from a theorem of Mathas on the action of $$T_{w_0}$$ on cell representations. The relevant paper is Mathas, On the Left Cell Representations of Iwahori‐Hecke Algebras of Finite Coxeter Groups. Proposition 3.17 is the relevant result.