Star operations on strings of adjacent simple reflections

Let $n\ge 3$ and $1\le i\le n-2$, and put $r=s_i$, $t=s_{i+1}$. These simple reflections satisfy $(rt{)}^3=1$. For $w\in S_n$, let $R(w):=\{s:\mathrm{\ell }(ws)<\mathrm{\ell }(w)\}$ be its right descent set, using the one-line convention and inversion length of Permutation Weyl group and inversion length. Define $D_i:=\{w\in S_n:\mathrm{\mid }R(w)\cap \{r,t\}\mathrm{\mid }=1\}\mathrm{.}$

The subgroup $P_i:=⟨r,t⟩$ permutes the entries in positions $i,i+1,i+2$. In each right coset $C=w{P}_i$, let $a<b<c$ be the three entries in those positions, and let $\tilde{w}$ be the unique member whose entries there are $a,b,c$ in increasing order. Then $C$ consists of $\tilde{w},\tilde{w}r,\tilde{w}t,\tilde{w}rt,\tilde{w}tr,\tilde{w}rtr$, with lengths $\mathrm{\ell }(\tilde{w}),\mathrm{\ell }(\tilde{w})+1,\mathrm{\ell }(\tilde{w})+1,\mathrm{\ell }(\tilde{w})+2,\mathrm{\ell }(\tilde{w})+2,\mathrm{\ell }(\tilde{w})+3$, respectively. Its intersection with $D_i$ is the four middle elements.

Right multiplying by $r$ and $t$ swaps the first two and last two block entries; the six words $1,r,t,rt,tr,rtr$ give the six distinct reorderings, so $P_i\cong S_3$ and the displayed list exhausts $C$. Sorting gives the unique member with no internal inversions. Reordering the block does not change the total number of inversions involving a position outside it: an outside position lies either before all three entries or after all three, so its comparisons with the block depend only on the set $\{a,b,c\}$. The internal inversion counts of $abc,bac,acb,bca,cab,cba$ are $0,1,1,2,2,3$. Their right descent sets restricted to $\{r,t\}$ are respectively $\mathrm{\varnothing },\{r\},\{t\},\{t\},\{r\},\{r,t\}$, so precisely the four length-one and length-two elements lie in $D_i$.

The right star operation $w\mapsto w^{\ast }$ on $D_i$ is defined on each coset by

Thus $w\mapsto w^{\ast }$ is an involution of $D_i$. The left star operation is ${}^{\ast }w:=((w^{-1}{)}^{\ast }{)}^{-1}$ on $D_i^{-1}:=\{w^{-1}:w\in D_i\}$; it is also an involution.

In one-line notation, the sorted triple for $\tilde{w}$ is $abc$ with $a<b<c$. The four elements of $D_i$ have triples $bac$, $bca$, $acb$, $cab$, and the right star operation exchanges $bac\leftrightarrow bca$ and $acb\leftrightarrow cab$. Hence for every $w\in D_i$, the words $w$ and $w^{\ast }$ differ by exactly one elementary Knuth move in positions $i,i+1,i+2$ as defined in Knuth and dual Knuth equivalence for permutations.