Parabolic Subgroups and Double Coset Geometry

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.

Let $(S,m)$, $W$, $\mathrm{\ell }$ be as in Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, let $I,J\subseteq S$, and let $W_I$, $W^J$, ${}^{I}W$, ${}^I{W}^J$ be as defined there; for $d\in W$ write

$\mathrm{\Omega }(d):=W_{I}d{W}_J=\{udv:u\in W_I,v\in W_J\}$

for the double coset of $d$.

(1) Descent reduction. Fix a total order on the finite set $S$. Every set $\mathrm{\Omega }(d)$ contains an element of ${}^I{W}^J$. More precisely, starting from $x:=d$ and repeatedly replacing the current element $x$ by $sx$, where $s$ is the least element of $I$ with $\mathrm{\ell }(sx)<\mathrm{\ell }(x)$, if such an $s$ exists, and otherwise by $xs$, where $s$ is the least element of $J$ with $\mathrm{\ell }(xs)<\mathrm{\ell }(x)$, if such an $s$ exists, the process stops after at most $\mathrm{\ell }(d)$ replacements at an element of $\mathrm{\Omega }(d)\cap {}^I{W}^J$.

(2) Elements of ${}^I{W}^J$ have minimum length. Let $d\in {}^I{W}^J$. Then

$\mathrm{\ell }(d)\le \mathrm{\ell }(x)\text{for every }x\in \mathrm{\Omega }(d),\text{and}\mathrm{\ell }(x)=\mathrm{\ell }(d)⟺x=d\mathrm{.}$

Consequently every double coset $W_{I}w{W}_J$ contains exactly one element of ${}^I{W}^J$, and it is the unique element of minimum length of that double coset; conversely, every element of minimum length in its double coset $W_{I}x{W}_J$ lies in ${}^I{W}^J$.

(3) Additive factorization. Let $d\in {}^I{W}^J$ and $x\in \mathrm{\Omega }(d)$. Then there exist $u\in W_I$ and $v\in W_J$ with

$x=udv,\mathrm{\ell }(x)=\mathrm{\ell }(u)+\mathrm{\ell }(d)+\mathrm{\ell }(v)\mathrm{.}$

• The deletion step is Tits deletion, not a cancellation of equal letters: the theorem of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action provides positions $i<j$ whose deletion preserves the value of a non-reduced word, and the two deleted letters can be distinct simple reflections. The argument uses only the positions.
• The argument is choice-free: each deletion is the lexicographically least admissible pair of a finite nonempty set of positions, and the minima of double cosets are minima of nonempty subsets of $\mathbb{N}$.

Let $(S,m)$, $W$, $\mathrm{\ell }$ and $I,J\subseteq S$ be as in Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups, so that $W_I$, $W^I$, ${}^{I}W$ and the descents $D_L,D_R$ have the meaning fixed there. Let $V={\mathbb{R}}^S$ with its Coxeter form $B$ and positive cone $V_{+}=\{{\sum }_{s\in S}{\lambda }_s{e}_s:{\lambda }_s\ge 0\}$ (The real Coxeter form, its radical, reflections, and form-preserving maps), let $\rho :W\to \mathrm{G}\mathrm{L}(V)$ be the canonical reflection homomorphism with root system $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots (2)), and for $I\subseteq S$ put

$V_I:=\mathrm{span}\{e_s:s\in I\},{\mathrm{\Phi }}_I:=\{\rho (w)e_s:w\in W_I,s\in I\}$

(Linear combination of a finite list, and the span $\mathrm{span}(S)$ as the smallest linear subspace containing $S$); here $V_{\mathrm{\varnothing }}=\{0\}$ and ${\mathrm{\Phi }}_{\mathrm{\varnothing }}=\mathrm{\varnothing }$.

(1) Intersections. For all $I,J\subseteq S$,

$W_I\cap W_J=W_{I\cap J}\mathrm{.}$

(2) Parabolic root subsystems. For every $I\subseteq S$ one has $\rho (w)V_I=V_I$ for all $w\in W_I$; consequently

${\mathrm{\Phi }}_I=\mathrm{\Phi }\cap V_I=\{\alpha \in \mathrm{\Phi }:\alpha \text{ is a real linear combination of the simple roots }{e}_s,s\in I\}\mathrm{.}$

Moreover ${\mathrm{\Phi }}_I={\mathrm{\Phi }}_I^{+}\bigsqcup {\mathrm{\Phi }}_I^{-}$ with ${\mathrm{\Phi }}_I^{\pm }:={\mathrm{\Phi }}_I\cap {\mathrm{\Phi }}^{\pm }={\mathrm{\Phi }}^{\pm }\cap V_I$, and the reflections lying in $W_I$ are exactly the elements $t_{\alpha }\in T$ with $\alpha \in {\mathrm{\Phi }}_I^{+}$ in the root-reflection dictionary of The inversion formula $\mathrm{\mid }N(w)\mathrm{\mid }=\mathrm{\ell }(w)$, the root-reflection dictionary and strong exchange (1).

(3) Coset minima are global minima. Let $I\subseteq S$, $w\in W$, and let $d\in W^I$ be the unique element of $W^I\cap w{W}_I$, i.e. the unique $d\in W^I$ with $w\in d{W}_I$ (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (2)). Then

$\mathrm{\ell }(d)\le \mathrm{\ell }(x)\text{for every }x\in w{W}_I,\text{and}\mathrm{\ell }(d)=\mathrm{\ell }(x)⟺x=d;$

equivalently

$W^I=\{d\in W:\mathrm{\ell }(d)\le \mathrm{\ell }(dv)\text{ for all }v\in W_I\},{}^{I}W=\{d\in W:\mathrm{\ell }(d)\le \mathrm{\ell }(vd)\text{ for all }v\in W_I\}\mathrm{.}$

Thus each set $w{W}_I$ contains exactly one element of $W^I$, namely its unique element of minimum length, and symmetrically each right coset $W_{I}w$ contains exactly one element of ${}^{I}W$, namely its unique element of minimum length.