Left and right coset minima and a double coset decomposition in S4

The two parabolics and their intersection. By [F1] the group $W_I=⟨s_1,s_2⟩$ is the type-$A_2$ Coxeter group, of order $6$, consisting of the permutations of $\{1,2,3\}$ extended by $4\mapsto 4$, i.e. the permutations with $w(4)=4$: the six elements are $1234$, $s_1=2134$, $s_2=1324$, $s_1{s}_2=2314$, $s_2{s}_1=3124$ and $s_1{s}_2{s}_1=s_2{s}_1{s}_2=3214$. Likewise $W_J=⟨s_2,s_3⟩$ consists of the permutations with $w(1)=1$: $1234$, $s_3=1243$, $s_2=1324$, $s_2{s}_3=1342$, $s_3{s}_2=1423$ and $s_2{s}_3{s}_2=1432$. Their intersection is $\{1234,1324\}=W_{\{s_2\}}=W_{I\cap J}$ by [F3].

The quotients and the two double cosets. Right multiplication by $s_i$ swaps the entries of the one-line form in positions $i$ and $i+1$, so it changes the inversion number by $\pm 1$, and decreases it exactly when those two entries are in decreasing order, that is when $w(i)>w(i+1)$; by [F1] the same holds for $\mathrm{\ell }$. Left multiplication by $s_i$ swaps the values $i$ and $i+1$ wherever they occur, so it decreases the inversion number exactly when the value $i$ occurs after the value $i+1$, that is when $w^{-1}(i)>w^{-1}(i+1)$. Hence $w\in W^J$ iff $w(2)<w(3)$ and $w(3)<w(4)$, which for the $24$ permutations gives $W^J=\{1234,2134,3124,4123\}$, and $w\in {}^{I}W$ iff $w^{-1}(1)<w^{-1}(2)$ and $w^{-1}(2)<w^{-1}(3)$, which gives ${}^{I}W=\{1234,1243,1423,4123\}$. Intersecting, ${}^I{W}^J=\{1234,4123\}$. By [F4] each double coset contains exactly one element of ${}^I{W}^J$, so the two double cosets $W_{I}1234W_J$ and $W_{I}4123W_J$ are distinct and their union is all of $S_4$.

The second pair of parabolics. Now $I=\{s_1\}$, $J=\{s_3\}$, so $W_I=\{1234,2134\}$ and $W_J=\{1234,1243\}$. An element lies in ${}^{I}W$ iff $w^{-1}(1)<w^{-1}(2)$ (the value $1$ occurs before the value $2$), and in $W^J$ iff $w(3)<w(4)$; listing the $24$ permutations gives ${}^I{W}^J=\{1234,1324,1423,3124,3412,4123,4312\}$, seven elements, hence seven double cosets by [F4]. Computing each double coset $W_{I}d{W}_J$ by multiplying the two generators on the left and right gives sizes $4,4,4,4,4,2,2$ for the minima $1234,1324,1423,3124,4123,3412,4312$; for the last two minima $d=3412$ and $d=4312$ one computes $d^{-1}{s}_{1}d=s_3$ in both cases, so $K=\{s_1\}$ and $W_I^K=\{1\}$ there, while for the first five minima $d^{-1}{s}_{1}d\mathrm{\notin }\{s_3\}$, so $K=\mathrm{\varnothing }$ and $W_I^K=\{1,s_1\}$; the sizes agree with $\mathrm{\mid }{W}_I^K\mathrm{\mid }\cdot \mathrm{\mid }{W}_J\mathrm{\mid }=4$ and $2$. Explicitly, $W_{I}3412W_J=\{3412,3421\}$ and $W_{I}4123W_J=\{4123,4132,4213,4231\}$.

The two double cosets of part (iii). For the pair $I=\{s_1,s_2\}$, $J=\{s_2,s_3\}$, listing the products $udv$ with $u\in W_I$, $v\in W_J$ gives $W_{I}1234W_J$ of order $18$ and $W_{I}4123W_J$ of order $6$, with $18+6=24$; by step 1.2 these two sets cover $S_4$ and are disjoint, and by [F4] their minima are $1234$ and $4123$. For $d=1234$ one has $d^{-1}{s}_{i}d=s_i$, so $K=\{s_1,s_2\}\cap \{s_2,s_3\}=\{s_2\}$, and the right-descent test $\mathrm{\ell }(u{s}_2)>\mathrm{\ell }(u)$ selects $W_I^K=\{1234,2134,3124\}$ from $W_I$; for $d=4123$ one computes $d^{-1}{s}_{1}d=s_2$ and $d^{-1}{s}_{2}d=s_3$, both of which lie in $J$, so $K=\{s_1,s_2\}$ and $W_I^K=\{1234\}$. The sizes match $\mathrm{\mid }{W}_I^K\mathrm{\mid }\cdot \mathrm{\mid }{W}_J\mathrm{\mid }=3\cdot 6=18$ and $1\cdot 6=6$, and the finite formula gives $18=6\cdot 6\mathrm{/}2$ and $6=6\cdot 6\mathrm{/}6$, since $\mathrm{\mid }{W}_K\mathrm{\mid }=\mathrm{\mid }\{1,s_2\}\mathrm{\mid }=2$ and $\mathrm{\mid }{W}_K\mathrm{\mid }=\mathrm{\mid }{W}_I\mathrm{\mid }=6$ respectively.

Coset minima and the normal form of $2143$. Let $w=2143$, of length $2$ by [F1]. Multiplying the six elements of $W_I$ into $w$ gives $W_{I}w=\{1243,1342,2143,2341,3142,3241\}$, whose element of least length is $1243=s_3$, of length $1$, and this is the unique element of ${}^{I}W$ in the right coset by step 1.2; multiplying the six elements of $W_J$ on the right gives $w{W}_J=\{2134,2143,2314,2341,2413,2431\}$, whose element of least length is $2134=s_1$, of length $1$, the unique element of $W^J$ in the left coset by step 1.2. Since $2143=2134\cdot 1243$ and $2134\in W_I$, $1243\in W_J$, the element $2143$ lies in the double coset of $d=1234$, whose minimum is $1234$ by step 2.1; the representation in the normal form of [F4] for $K=\{s_2\}$ is $2143=2134\cdot 1234\cdot 1243$, with $2134\in W_I^K=\{1234,2134,3124\}$ by step 2.1 and $1243\in W_J$, and the lengths add: $\mathrm{\ell }(2134)+\mathrm{\ell }(1234)+\mathrm{\ell }(1243)=1+0+1=2=\mathrm{\ell }(2143)$, in agreement with [F4] and [F6]. This completes the example.