Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups

Let $(S,m)$ be a finite Coxeter matrix, let $W$ be the presented group with length function $\mathrm{\ell }$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), and let $T=\{ws{w}^{-1}:w\in W,s\in S\}$ be the set of reflections of $W$ (The canonical reflection homomorphism, roots, reflections, and the positive cone). Throughout, $I$ and $J$ denote subsets of $S$.

(1) Standard parabolic subgroups. For $I\subseteq S$ put

$W_I:=⟨s:s\in I⟩\le W,$

the subgroup generated by $I$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups); $W_I$ is the standard parabolic subgroup of $(W,S)$ of type $I$. Thus $W_{\mathrm{\varnothing }}=\{1\}$ and $W_S=W$. By Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1),(2): for every $w\in W$ one has $w\in W_I$ if and only if $S(w)\subseteq I$, where $S(w)$ is the (well-defined) set of letters of any reduced expression of $w$; $(W_I,I)$ is a Coxeter system; its intrinsic length function agrees with $\mathrm{\ell }{\mathrm{\mid }}_{W_I}$; and $W_I\cap S=I$.

(2) Descents and the one-sided quotients. For $w\in W$ put

$D_L(w):=\{s\in S:\mathrm{\ell }(sw)<\mathrm{\ell }(w)\},D_R(w):=\{s\in S:\mathrm{\ell }(ws)<\mathrm{\ell }(w)\};$

each of $\mathrm{\ell }(sw)-\mathrm{\ell }(w)$ and $\mathrm{\ell }(ws)-\mathrm{\ell }(w)$ lies in $\{\pm 1\}$ (Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action (1)). Define

$W^I:=\{w\in W:\mathrm{\ell }(ws)>\mathrm{\ell }(w)\text{ for all }s\in I\}=\{w\in W:D_R(w)\cap I=\mathrm{\varnothing }\},$

${}^{I}W:=\{w\in W:\mathrm{\ell }(sw)>\mathrm{\ell }(w)\text{ for all }s\in I\}=\{w\in W:D_L(w)\cap I=\mathrm{\varnothing }\}\mathrm{.}$

Conventions are fixed here: in ${}^{I}W$ the subset $I$ is tested on the left, in $W^I$ it is tested on the right; $\mathrm{\ell }(sw)$ means left multiplication by $s$ and $\mathrm{\ell }(ws)$ right multiplication. The interpretation of these sets as sets of coset representatives is not part of the definition; it is quoted from Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3) and re-examined in Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives ↗ (3): every $w\in W$ has a unique factorization

$w=ud(u\in W_I,d\in {}^{I}W),\mathrm{\ell }(w)=\mathrm{\ell }(u)+\mathrm{\ell }(d),$

where $d$ is the unique element of minimal length in the right coset $W_{I}w$, and it satisfies $\mathrm{\ell }(ud)=\mathrm{\ell }(u)+\mathrm{\ell }(d)$ for all $u\in W_I$; symmetrically every $w\in W$ has a unique factorization

$w=dv(d\in W^I,v\in W_I),\mathrm{\ell }(w)=\mathrm{\ell }(d)+\mathrm{\ell }(v),$

where $d$ is the unique element of minimal length in the left coset $w{W}_I$, and it satisfies $\mathrm{\ell }(dv)=\mathrm{\ell }(d)+\mathrm{\ell }(v)$ for all $v\in W_I$. Thus ${}^{I}W$ consists exactly of the elements of minimal length in the right cosets $W_{I}w=\{uw:u\in W_I\}$ and $W^I$ consists exactly of the elements of minimal length in the sets $w{W}_I=\{wv:v\in W_I\}$. Inversion gives $W^I=({}^{I}W{)}^{-1}$, because $\mathrm{\ell }(x)=\mathrm{\ell }(x^{-1})$ and $(sw{)}^{-1}=w^{-1}s$.

(3) The two-sided quotients. Define

${}^I{W}^J:={}^{I}W\cap W^J=\{w\in W:\mathrm{\ell }(sw)>\mathrm{\ell }(w)\text{ for all }s\in I\text{ and }\mathrm{\ell }(ws)>\mathrm{\ell }(w)\text{ for all }s\in J\};$

thus ${}^{\mathrm{\varnothing }}{W}^J=W^J$, ${}^I{W}^{\mathrm{\varnothing }}={}^{I}W$ and ${}^{\mathrm{\varnothing }}{W}^{\mathrm{\varnothing }}=W$, and the members of ${}^I{W}^J$ are exactly the elements of $W$ with no left descent in $I$ and no right descent in $J$. Whether ${}^I{W}^J$ is a set of representatives of the double cosets $W_{I}w{W}_J$ is again a theorem, not a definition; it is proved in Unique minimal double coset representatives and the additive normal form u-d-v ↗.

(4) Parabolic subgroups and reflection subgroups. A subgroup $H\le W$ is a parabolic subgroup of $(W,S)$ if

$H=w{W}_I{w}^{-1}\text{for some }w\in W,I\subseteq S;$

a parabolic subgroup with $w=1$ is standard. A subgroup $H\le W$ is a reflection subgroup if $H=⟨H\cap T⟩$, i.e. if $H$ is generated by the reflections it contains. Every parabolic subgroup is a reflection subgroup, since $w{W}_I{w}^{-1}=⟨ws{w}^{-1}:s\in I⟩$ and $ws{w}^{-1}\in T$ for all $s\in S$, $w\in W$. No converse is asserted: it is not claimed here that a subgroup generated by reflections is parabolic, and the companion examples page computes test cases for the two distinctions created by this definition --- standard parabolic versus parabolic, and reflection subgroup versus parabolic subgroup.

• Well-definedness. $W_I$ is the subgroup generated by a subset of the group $W$ and exists by The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups; $W_{\mathrm{\varnothing }}=\{1\}$ is the trivial subgroup and $W_S=W$. The sets $W^I$, ${}^{I}W$ and ${}^I{W}^J$ are subsets of $W$ specified by a predicate and are therefore well-defined sets. The two displayed factorizations of (2) and the description of $W^I$, ${}^{I}W$ as sets of minimal representatives are quoted from Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)-(3), which proves them from the deletion and exchange properties before they are used here; the inversion identity $W^I=({}^{I}W{)}^{-1}$ uses $\mathrm{\ell }(x)=\mathrm{\ell }(x^{-1})$, $(sw{)}^{-1}=w^{-1}s$ (Group and abelian group) and the equivalence $\mathrm{\ell }(sw)<\mathrm{\ell }(w)\iff \mathrm{\ell }(w^{-1}s)<\mathrm{\ell }(w^{-1})$. The only genuinely two-sided assertion is deferred: this item introduces ${}^I{W}^J$ as a set of descent-free elements and asserts nothing about double cosets; the property that every double coset $W_{I}w{W}_J$ contains exactly one element of ${}^I{W}^J$, and that this element is its minimum, is established in Unique minimal double coset representatives and the additive normal form u-d-v ↗, the later proof that justifies the definition for its applications.
• Left and right conventions are not symmetric by accident. In ${}^{I}W$ the descent is tested by left multiplication, in $W^I$ by right multiplication; the right coset $W_{I}w$ is represented by ${}^{I}W$ and the left coset $w{W}_I$ by $W^I$. Reversing either convention produces false statements already in $S_3$ with $I=\{s_1\}$ (for example $s_2{s}_1$ has no factorization $ud$ with $d\in W^I$), which is why they are recorded here once and used consistently.
• No converse and no Choice. The inclusion "parabolic subgroups are reflection subgroups" in (4) is immediate from the definitions, since a conjugate $ws{w}^{-1}$ of a simple reflection is a reflection; nothing is asserted in the other direction. No use of the Axiom of Choice is made: all constructions are subgroups and subsets of the fixed group $W$, and the minima in (2) are minima of nonempty subsets of the natural numbers.