Parabolic subgroup of a reflection group

In a Euclidean space (such as the Euclidean plane, ordinary three-dimensional space, or their higher-dimensional analogues), a reflection is a symmetry of the space across a mirror (technically, across a subspace of dimension one smaller than the whole space) that fixes the vectors that lie on the mirror and sends the vectors orthogonal to the mirror to their negatives. A finite real reflection group {{mvar|W}} is a finite group generated by reflections (that is, every linear transformation in {{mvar|W}} is a composition of some of the reflections in {{mvar|W}}).{{sfnp|Kane|2001|p=1}} For example, the symmetries of a regular polygon in the plane form a reflection group (called the dihedral group), because each rotation symmetry of the polygon is a composition of two reflections.{{sfnp|Kane|2001|pp=8–14}} Finite real reflection groups can be generalized in various ways,{{sfnp|Humphreys|1990|pp=xi–xii}} and the definition of parabolic subgroup depends on the choice of definition.

Each finite real reflection group {{mvar|W}} has the structure of a Coxeter group:{{sfnp|Kane|2001|p=1}} this means that {{mvar|W}} contains a subset {{mvar|S}} of reflections (called simple reflections) such that {{mvar|S}} generates {{mvar|W}}, subject to relations of the form

$$W\; =\; \backslash langle\; S\; \backslash mid\; (s\; s\text{'})^\{m\_\{s,\; s\text{'}\}\}\; =\; 1\; \backslash rangle,$$

where {{math|1}} denotes the identity in {{mvar|W}} and the $m\_\{s,\; s\text{'}\}$ are numbers that satisfy $m\_\{s,\; s\}\; =\; 1$ for $s\; \backslash in\; S$ and $m\_\{s,\; s\text{'}\}\; \backslash in\; \backslash \{2,\; 3,\; \backslash ldots\backslash \}$ for $s\; \backslash neq\; s\text{'}\; \backslash in\; S$.{{efn|In general Coxeter groups, the possibility $m\_\{s,\; s\text{'}\}\; =\; \backslash infty$ is also allowed, meaning that no relation holds between {{mvar|s}} and {{mvar|s}}{{'}}—but this situation cannot occur in a finite group.}}{{sfnp|Humphreys|1990|loc=§1.9}} Thus, the Coxeter groups form one generalization of finite real reflection groups.

A separate generalization is to consider the geometric action on vector spaces whose underlying field is not the real numbers.{{sfnp|Kane|2001|p=1}} Especially, if one replaces the real numbers with the complex numbers, with a corresponding generalization of the notion of a reflection, one arrives at the definition of a complex reflection group.{{efn|Such groups are also known as unitary reflection groups or complex pseudo-reflection groups in some sources. Similarly, sometimes complex reflections (linear transformations that fix a hyperplane pointwise) are called pseudo-reflections.{{sfnp|Kane|2001|p=160}}}} Every real reflection group can be complexified to give a complex reflection group, so the complex reflection groups form another generalization of finite real reflection groups.{{sfnp|Lehrer|Taylor|2009|p=1}}{{sfnp|Humphreys|1990|p = 66}}

File:Standard parabolic subgroups of the symmetric group S 4.svgs), with identity element {{mvar|ι}}]]

Suppose that {{mvar|W}} is a Coxeter group with a finite set {{mvar|S}} of simple reflections. For each subset {{mvar|I}} of {{mvar|S}}, let $W\_I$ denote the subgroup of {{mvar|W}} generated by $I$. Such subgroups are called standard parabolic subgroups of the Coxeter system $(W,\; S)$.{{sfnp|Björner|Brenti|2005|loc = §2.4}}{{sfnp|Humphreys|1990|loc=§5.5}} In the extreme cases, $W\_\backslash varnothing$ is the trivial subgroup (containing just the identity element of {{mvar|W}}) and {{nowrap|$W\_S\; =\; W$.{{sfnp|Humphreys|1990|loc=§1.10}}}}

The pair $(W\_I,\; I)$ is again a Coxeter system. Moreover, the Coxeter group structure on $W\_I$ is compatible with that on {{mvar|W}}, in the following sense: if $\backslash ell\_S$ denotes the length function on {{mvar|W}} with respect to {{mvar|S}} (so that $\backslash ell\_S(w)\; =\; k$ if the element {{mvar|w}} of {{mvar|W}} can be written as a product of {{mvar|k}} elements of {{mvar|S}} and not fewer), then for every element {{mvar|w}} of $W\_I$, one has that $\backslash ell\_S(w)\; =\; \backslash ell\_I(w)$. That is, the length of {{mvar|w}} is the same whether it is viewed as an element of {{mvar|W}} or of $W\_I$.{{sfnp|Björner|Brenti|2005|loc = §2.4}}{{sfnp|Humphreys|1990|loc=§5.5}} The same is true of the Bruhat order: if {{mvar|u}} and {{mvar|w}} are elements of $W\_I$, then $u\; \backslash leq\; w$ in the Bruhat order on $W\_I$ if and only if $u\; \backslash leq\; w$ in the Bruhat order on {{mvar|W}}.{{sfnp|Humphreys|1990|loc=§5.10}}

If {{mvar|I}} and {{mvar|J}} are two subsets of {{mvar|S}}, then $W\_I\; =\; W\_J$ if and only if $I\; =\; J$. Also, $W\_I\; \backslash cap\; W\_J\; =\; W\_\{I\backslash cap\; J\}$, and the smallest group $\backslash langle\; W\_I,\; W\_J\; \backslash rangle$ that contains both $W\_I$ and $W\_J$ is $W\_\{I\; \backslash cup\; J\}$. Consequently, the lattice of standard parabolic subgroups of {{mvar|W}} is a Boolean lattice.{{sfnp|Björner|Brenti|2005|loc = §2.4}}{{sfnp|Humphreys|1990|loc=§5.5}}

Given a standard parabolic subgroup $W\_I$ of a Coxeter system $(W,\; S)$, the cosets of $W\_I$ in {{mvar|W}} have a particularly nice system of representatives: let $W^I$ denote the set

$$W^I\; =\; \backslash \{\; w\; \backslash in\; W\; \backslash colon\; \backslash ell\_S(ws)\; >\; \backslash ell\_S(w)\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; I\backslash \}$$

of elements in {{mvar|W}} that do not have any element of {{mvar|I}} as a right descent.{{efn|A right descent of an element {{mvar|w}} in a Coxeter group is a simple reflection {{mvar|s}} such that .{{sfnp|Björner|Brenti|2005|p=17}}}} Then for each $w\; \backslash in\; W$, there are unique elements $u\; \backslash in\; W^I$ and $v\; \backslash in\; W\_I$ such that $w\; =\; uv$. Moreover, this is a length-additive product, that is, $\backslash ell\_S(w)\; =\; \backslash ell\_S(u)\; +\; \backslash ell\_S(v)$. Furthermore, {{mvar|u}} is the element of minimum length in the coset $w\; W\_I$.{{sfnp|Björner|Brenti|2005|loc = §2.4}}{{sfnp|Humphreys|1990|loc=§5.12}} An analogous construction is valid for right cosets.{{sfnp|Björner|Brenti|2005|p=41}} The collection of all left cosets of standard parabolic subgroups is one possible construction of the Coxeter complex.{{sfnp|Björner|Brenti|2005|pp=86–7}}

In terms of the Coxeter–Dynkin diagram, the standard parabolic subgroups arise by taking a subset of the nodes of the diagram and the edges induced between those nodes, erasing all others.{{sfnp|Björner|Brenti|2005|p=39}} The only normal parabolic subgroups arise by taking a union of connected components of the diagram, and the whole group {{mvar|W}} is the direct product of the irreducible Coxeter groups that correspond to the components.{{sfnp|Humphreys|1990|pp=118, 129}}

File:Parabolic subgroups of a dihedral group.svg

Suppose that {{mvar|W}} is a complex reflection group acting on a complex vector space {{mvar|V}}. For any subset $A\; \backslash subseteq\; V$, let

$$W\_A\; =\; \backslash \{\; w\; \backslash in\; W\; \backslash colon\; w(a)\; =\; a\; \backslash text\{\; for\; all\; \}\; a\; \backslash in\; A\backslash \}$$

be the subset of {{mvar|W}} consisting of those elements in {{mvar|W}} that fix each element of {{mvar|A}}.{{efn|Sometimes such subgroups are called isotropy groups.{{sfnp|Kane|2001|p = 60}}}} Such a subgroup is called a parabolic subgroup of {{mvar|W}}.{{sfnp|Lehrer|Taylor|2009|p=171}} In the extreme cases, $W\_\{\backslash varnothing\}\; =\; W\_\{\backslash \{0\backslash \}\}\; =\; W$ and $W\_V$ is the trivial subgroup of {{mvar|W}} that contains only the identity element.

It follows from a theorem of {{harvtxt|Steinberg|1964}} that each parabolic subgroup $W\_A$ of a complex reflection group {{mvar|W}} is a reflection group, generated by the reflections in {{mvar|W}} that fix every point in {{mvar|A}}.{{sfnp|Lehrer|Taylor|2009|loc = §9.7}} Since {{mvar|W}} acts linearly on {{mvar|V}}, $W\_A\; =\; W\_\{\backslash overline\{A\}\}$ where $\backslash overline\{A\}$ is the span of {{mvar|A}} (that is, the smallest linear subspace of {{mvar|V}} that contains {{mvar|A}}).{{sfnp|Lehrer|Taylor|2009|p=171}} In fact, there is a simple choice of subspaces {{mvar|A}} that index the parabolic subgroups: each reflection in {{mvar|W}} fixes a hyperplane (that is, a subspace of {{mvar|V}} whose dimension is {{math|1}} less than that of {{mvar|V}}) pointwise, and the collection of all these hyperplanes is the reflection arrangement of {{mvar|W}}.{{sfnp|Orlik|Terao|1992|p=215}} The collection of all intersections of subsets of these hyperplanes,{{efn|Including the entire space {{mvar|V}}, as the empty intersection.}} partially ordered by inclusion, is a lattice $L\_W$.{{sfnp|Orlik|Terao|1992|loc=§2.1}} The elements of the lattice are precisely the fixed spaces of the elements of {{mvar|W}} (that is, for each intersection {{mvar|I}} of reflecting hyperplanes, there is an element $w\; \backslash in\; W$ such that $\backslash \{v\; \backslash in\; V\; \backslash colon\; w(v)\; =\; v\backslash \}\; =\; I$).{{sfnp|Lehrer|Taylor|2009|loc=§9.3}}{{sfnp|Broué|2010|loc = §4.2.4}} The map that sends $I\; \backslash mapsto\; W\_I$ for $I\; \backslash in\; L\_W$ is an order-reversing bijection between subspaces in $L\_W$ and parabolic subgroups of {{mvar|W}}.{{sfnp|Broué|2010|loc = §4.2.4}}

Let {{mvar|W}} be a finite real reflection group; that is, {{mvar|W}} is a finite group of linear transformations on a finite-dimensional real Euclidean space that is generated by orthogonal reflections. As mentioned above (see {{slink||Background: reflection groups}}), {{mvar|W}} may be viewed as both a Coxeter group and as a complex reflection group in a natural way. For a real reflection group {{mvar|W}}, the parabolic subgroups of {{mvar|W}} (viewed as a complex reflection group) are not all standard parabolic subgroups of {{mvar|W}} (when viewed as a Coxeter group, after specifying a fixed Coxeter generating set {{mvar|S}}), as there are many more subspaces in the intersection lattice of its reflection arrangement than subsets of {{mvar|S}}. However, in a finite real reflection group {{mvar|W}}, every parabolic subgroup is conjugate to a standard parabolic subgroup with respect to {{mvar|S}}.{{sfnp|Kane|2001|loc=§5.2}}

File:Parabolic subgroups of the Weyl group B 2.svg

The symmetric group $S\_n$, which consists of all permutations of $\backslash \{1,\; \backslash ldots,\; n\backslash \}$, is a Coxeter group with respect to the set of adjacent transpositions $(1\backslash \; 2)$, ..., $(n\; -\; 1\backslash \; n)$. The standard parabolic subgroups of $S\_n$ (which are also known as Young subgroups) are the subgroups of the form $S\_\{a\_1\}\; \backslash times\; \backslash cdots\; \backslash times\; S\_\{a\_k\}$, where $a\_1,\; \backslash ldots,\; a\_k$ are positive integers with sum {{mvar|n}}, in which the first factor in the direct product permutes the elements $\backslash \{1,\; \backslash ldots,\; a\_1\backslash \}$ among themselves, the second factor permutes the elements $\backslash \{a\_1\; +\; 1,\; \backslash ldots,\; a\_1\; +\; a\_2\backslash \}$ among themselves, and so on.{{sfnp|Kane|2001|p=58}}{{sfnp|Björner|Brenti|2005|p=41}}

The hyperoctahedral group $S^B\_n$, which consists of all signed permutations of $\backslash \{\backslash pm\; 1,\; \backslash ldots,\; \backslash pm\; n\backslash \}$ (that is, the bijections {{mvar|w}} on that set such that $w(-i)\; =\; -\; w(i)$ for all {{mvar|i}}), is a Coxeter group with respect to the generating set $\backslash \{(1\backslash \; -1),\; (1\backslash \; 2)\; (-1\backslash \; -2),\; \backslash ldots,\; (n\; -\; 1\backslash \; n)\; (-n+1\backslash \; -n)\backslash \}$. Its maximal standard parabolic subgroups are the setwise stabilizers of $\backslash \{i,\; \backslash ldots,\; n\backslash \}$ for $i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}$.{{sfnp|Björner|Brenti|2005|p=248}}

In $S^B\_2$, the maximal standard parabolic subgroups are $\backslash \{\backslash iota,\; (1\backslash \; -1)\backslash \}$ (stabilizing the set {{tmath| \{2\} }}) and $\backslash \{\backslash iota,\; (1\backslash \; 2)(-1\backslash \; -2)\backslash \}$ (stabilizing the set {{tmath|\{1, 2\} }}). Under conjugation, these yield the additional parabolic subgroups $\backslash \{\backslash iota,\; (2\backslash \; -2)\backslash \}$ and $\backslash \{\backslash iota,\; (1\backslash \; -2)(-1\backslash \; 2)\backslash \}$. The only other parabolic subgroups are the whole group and the trivial subgroup; when ordered by containment, these form a non-Boolean lattice, as depicted in the figure at right. (The hyperoctahedral group $S^B\_2$ is isomorphic to the dihedral group $D\_\{2\; \backslash times\; 4\}$ as Coxeter systems, which is why we get the same lattice of parabolic subgroups of the dihedral group $D\_\{2\; \backslash times\; 4\}$ in the previous figure.)

In a Coxeter system with a finite generating set {{mvar|S}} of simple reflections, one may define a parabolic subgroup to be any conjugate of a standard parabolic subgroup. Under this definition, it is still true that the intersection of any two parabolic subgroups is a parabolic subgroup. The same does not hold in general for Coxeter groups of infinite rank.{{sfnp|Nuida|2012}}

If {{mvar|W}} is a group and {{mvar|T}} is a subset of {{mvar|W}}, the pair $(W,\; T)$ is called a dual Coxeter system if there exists a subset {{mvar|S}} of {{mvar|T}} such that $(W,\; S)$ is a Coxeter system and

$$T\; =\; \backslash \{\; w\; s\; w^\{-1\}\; \backslash colon\; w\; \backslash in\; W,\; s\; \backslash in\; S\; \backslash \},$$

so that {{mvar|T}} is the set of all reflections (conjugates of the simple reflections) in {{mvar|W}}. For a dual Coxeter system $(W,\; T)$, a subgroup of {{mvar|W}} is said to be a parabolic subgroup if it is a standard parabolic subgroup (as in {{section link|#In Coxeter groups}}) of $(W,\; S)$ for some choice of simple reflections {{mvar|S}} for {{nowrap|$(W,\; T)$.{{sfnp|Baumeister|Dyer|Stump|Wegener|2014}}{{efn|In the case of a finite real reflection group, this definition differs from the classical one, where {{mvar|S}} necessarily comes from the reflections whose reflecting hyperplanes form the boundaries of a chamber.{{sfnp|Reiner|Ripoll|Stump|2017|loc=Example 1.2}}}}}}

In some dual Coxeter systems, all sets of simple reflections are conjugate to each other; in this case, the parabolic subgroups with respect to one simple system (that is, the conjugates of the standard parabolic subgroups) coincide with the parabolic subgroups with respect to any other simple system. However, even in finite examples, this may not hold: for example, if {{mvar|W}} is the dihedral group with {{math|10}} elements, viewed as symmetries of a regular pentagon, and {{mvar|T}} is the set of reflection symmetries of the polygon, then any pair of reflections in {{mvar|T}} forms a simple system for $(W,\; T)$, but not all pairs of reflections are conjugate to each other.{{sfnp|Baumeister|Dyer|Stump|Wegener|2014}} Nevertheless, if {{mvar|W}} is finite, then the parabolic subgroups (in the sense above) coincide with the parabolic subgroups in the classical sense (that is, the conjugates of the standard parabolic subgroups with respect to a single, fixed, choice of simple reflections {{mvar|S}}).{{sfnp|Baumeister|Gobet|Roberts|Wegener|2017|loc=Proposition 1.4 and Corollary 4.4}} The same result does not hold in general for infinite Coxeter groups.{{sfnp|Gobet|2017|loc=Example 2.2}}

When {{mvar|W}} is an affine Coxeter group, the associated finite Weyl group is always a maximal parabolic subgroup, whose Coxeter–Dynkin diagram is the result of removing one node from the diagram of {{mvar|W}}. In particular, the length functions on the finite and affine groups coincide.{{sfnp|Humphreys|1990|p=114}} In fact, every standard parabolic subgroup of an affine Coxeter group is finite.{{sfnp|Humphreys|1990|p=96}} As in the case of finite real reflection groups, when we consider the action of an affine Coxeter group {{mvar|W}} on a Euclidean space {{mvar|V}}, the conjugates of the standard parabolic subgroups of {{mvar|W}} are precisely the subgroups of the form

$$\backslash \{w\; \backslash in\; W\; \backslash colon\; w(a)\; =\; a\; \backslash text\{\; for\; all\; \}\; a\; \backslash in\; A\backslash \}$$

for some subset {{mvar|A}} of {{mvar|V}}.{{sfnp|Kane|2001|p=130}}

If {{mvar|W}} is a crystallographic Coxeter group,{{efn|That is, if {{mvar|W}} is a (possibly infinite) Coxeter group that stabilizes a lattice in its natural geometric representation.}} then every parabolic subgroup of {{mvar|W}} is also crystallographic.{{sfnp|Humphreys|1990|p=136}}

If {{mvar|G}} is an algebraic group and {{mvar|B}} is a Borel subgroup for {{mvar|G}}, then a parabolic subgroup of {{mvar|G}} is any subgroup that contains {{mvar|B}}.{{efn|This use of the phrase "parabolic subgroup" was introduced by Roger Godement in his paper {{harvtxt|Godement|1961}}.{{sfnp|Borel|2001|loc = chapter VI, section 2}}}} If furthermore {{mvar|G}} has a (B, N) pair, then the associated quotient group $W\; =\; B\; /\; (B\; \backslash cap\; N)$ is a Coxeter group, called the Weyl group of {{mvar|G}}. Then the group {{mvar|G}} has a Bruhat decomposition $G\; =\; \backslash bigsqcup\_\{w\; \backslash in\; W\}\; BwB$ into double cosets (where $\backslash sqcup$ is the disjoint union), and the parabolic subgroups of {{mvar|G}} containing {{mvar|B}} are precisely the subgroups of the form

$$P\_J\; =\; BW\_JB$$

where $W\_J$ is a standard parabolic subgroup of {{mvar|W}}.{{sfnp|Digne|Michel|1991|pp=19–21}}

Suppose {{mvar|W}} is a Coxeter group of finite rank (that is, the set {{mvar|S}} of simple generators is finite). Given any subset {{mvar|X}} of {{mvar|W}}, one may define the parabolic closure of {{mvar|X}} to be the intersection of all parabolic subgroups containing {{mvar|X}}. As mentioned above, in this case the intersection of any two parabolic subgroups of {{mvar|W}} is again a parabolic subgroup of {{mvar|W}}, and consequently the parabolic closure of {{mvar|X}} is a parabolic subgroup of {{mvar|W}}; in particular, it is the (unique) minimal parabolic subgroup of {{mvar|W}} containing {{mvar|X}}.{{sfnp|Nuida|2012}} The same analysis applies to complex reflection groups, where the parabolic closure of {{mvar|X}} is also the pointwise stabiliser of the space of fixed points of {{mvar|X}}.{{sfnp|Taylor|2012}} The same does not hold for Coxeter groups of infinite rank.{{sfnp|Nuida|2012}}

Each Coxeter group is associated to another group called its Artin–Tits group or generalized braid group, which is defined by omitting the relations $s^2\; =\; 1$ for each generator $s\; \backslash in\; S$ from its Coxeter presentation.{{efn|The name "generalized braid group" arises from the fact that, in the special case $W\; =\; S\_n$ is the symmetric group, the associated Artin–Tits group is the braid group on {{mvar|n}} strands.}}{{sfnp|McCammond|Sulway|2017}} Although generalized braid groups are not reflection groups, they inherit a notion of parabolic subgroups: a standard parabolic subgroup of a generalized braid group is a subgroup generated by a subset of the standard generating set {{mvar|S}}, and a parabolic subgroup is any subgroup conjugate to a standard parabolic.{{sfnp|Cumplido|Gebhardt|González-Meneses|Wiest|2019}}

A generalized braid group is said to be of spherical type if the associated Coxeter group is finite. If {{mvar|B}} is a generalized braid group of spherical type, then the intersection of any two parabolic subgroups of {{mvar|B}} is also a parabolic subgroup. Consequently, the parabolic subgroups of {{mvar|B}} form a lattice under inclusion.{{sfnp|Cumplido|Gebhardt|González-Meneses|Wiest|2019}}

For a finite real reflection group {{mvar|W}}, the associated generalized braid group may be defined in purely topological language, without referring to a particular group presentation.{{efn|In particular, the group {{mvar|W}} acts on the complement of the complexification of the arrangement of its reflecting hyperplanes; the generalized braid group of {{mvar|W}} is the fundamental group of the quotient of this space under the action of {{mvar|W}}.}} This definition naturally extends to finite complex reflection groups.{{sfnp|Broué|2010|loc = §4.2.5}} Parabolic subgroups can also be defined in this setting.{{sfnp|González-Meneses|Marin|2022}}

{{noteslist}}

{{reflist}}

• {{citation | last1= Baumeister| first1= Barbara | last2= Dyer | first2= Matthew | last3=Stump | first3= Christian | last4= Wegener | first4= Patrick | title = A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements | journal = Proceedings of the American Mathematical Society | series = Series B | volume = 1 | year = 2014 | issue= 13 | pages= 149–154| arxiv= 1402.2500 | doi = 10.1090/S2330-1511-2014-00017-1 | doi-access = free}}
• {{citation | last1 = Baumeister | first1= Barbara |last2= Gobet |first2= Thomas | last3= Roberts |first3= Kieran|last4 = Wegener |first4= Patrick |title=On the Hurwitz action in finite Coxeter groups | journal = Journal of Group Theory | volume = 20 | year = 2017 | issue = 1 |pages = 103–131| doi= 10.1515/jgth-2016-0025 | arxiv= 1512.04764 | s2cid= 44035800 }}
• {{citation |title=Combinatorics of Coxeter groups |last1=Björner |first1=Anders |author1-link=Anders Björner |s2cid=115235335 |last2=Brenti |first2=Francesco |publisher=Springer |year=2005 |isbn=978-3540-442387 |doi=10.1007/3-540-27596-7}}
• {{citation | last = Borel | first = Armand | author-link = Armand Borel | title = Essays in the history of Lie groups and algebraic groups | series = History of Mathematics | volume = 21 | publisher = American Mathematical Society and London Mathematical Society | year = 2001 | isbn = 0-8218-0288-7}}
• {{citation | last = Broué | first = Michel | author-link = Michel Broué | title = Introduction to Complex Reflection Groups and Their Braid Groups | series = Lecture Notes in Mathematics | volume = 1988 | publisher = Springer-Verlag | year = 2010 | isbn = 978-3-642-11174-7 | doi = 10.1007/978-3-642-11175-4}}
• {{citation | last1 = Cumplido | first1 = María | last2 = Gebhardt | first2 = Volker | last3 = González-Meneses | first3 = Juan | last4 = Wiest | first4 = Bert | title = On parabolic subgroups of Artin–Tits groups of spherical type | journal = Advances in Mathematics | volume = 352 | year = 2019 | pages = 572–610 | doi = 10.1016/j.aim.2019.06.010 | doi-access = free| arxiv = 1712.06727 }}
• {{citation | last1 = Digne | first1 = François | last2 = Michel | first2 = Jean | title = Representations of finite groups of Lie type | series = London Mathematical Society Student Texts | volume = 21 | publisher = Cambridge University Press | year = 1991 | isbn = 0-521-40117-8}}
• {{citation | last = Gobet |first = Thomas |title = On cycle decompositions in Coxeter groups | journal = Séminaire Lotharingien de Combinatoire | volume = 78B | year = 2017 | pages = Art. 45 |arxiv = 1611.03442 | url = https://www.emis.de/journals/SLC/wpapers/FPSAC2017/45%20Gobet.html}}
• {{citation |last = Godement | first = Roger |author-link = Roger Godement | title = Groupes linéaires algébriques sur un corps parfait | journal = Séminaire Bourbaki | volume = 13 | year = 1961 | url = https://www.numdam.org/item/?id=SB_1960-1961__6__11_0}}
• {{citation | last1 = González-Meneses | first1 = Juan | last2 = Marin | first2 = Ivan | title = Parabolic subgroups of complex braid groups I | arxiv = 2208.11938v1 | year = 2022}}
• {{citation |title=Reflection groups and Coxeter groups |last=Humphreys |first=James E. |author-link=James E. Humphreys |publisher=Cambridge University Press |year=1990 |isbn=0-521-37510-X |s2cid=121077209 |doi=10.1017/CBO9780511623646}}
• {{citation |title=Reflection groups and invariant theory |last=Kane |first=Richard |publisher=Springer-Verlag |series=CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC |year=2001 |isbn=0-387-98979-X |s2cid=119694827 |doi=10.1007/978-1-4757-3542-0}}
• {{Citation | last1=Lehrer | first1=Gustav I. | last2=Taylor | first2=Donald E. | title=Unitary reflection groups | publisher=Cambridge University Press | series=Australian Mathematical Society Lecture Series | isbn=978-0-521-74989-3 | year=2009 | volume=20}}
• {{citation | last1 = McCammond | first1 = Jon | last2= Sulway | first2 = Robert | title = Artin groups of Euclidean type | journal = Inventiones Mathematicae | year = 2017 | volume = 210 | issue = 1 | pages = 231–282 | doi = 10.1007/s00222-017-0728-2| bibcode = 2017InMat.210..231M | s2cid = 253738806 | arxiv = 1312.7770 }}
• {{citation | last = Nuida | first = Koji | title = Locally parabolic subgroups in Coxeter groups of arbitrary ranks | journal = Journal of Algebra | volume = 350 | year = 2012 | pages = 207–217| doi = 10.1016/j.jalgebra.2011.11.005 | arxiv = 1006.4709 }}
• {{citation | title = Arrangements of Hyperplanes | last1 = Orlik | first1 = Peter | author1-link = Peter Orlik | last2 = Terao | first2 = Hiroaki | author2-link=Hiroaki Terao| publisher = Springer | isbn = 978-3-540-55259-8 | series = Grundlehren der mathematischen Wissenschaften | year = 1992 | doi=10.1007/978-3-662-02772-1}}
• {{citation | first1 = Victor | last1 = Reiner | first2= Vivien |last2 = Ripoll | first3= Christian | last3 = Stump | title= On non-conjugate Coxeter elements in well-generated reflection groups | journal = Mathematische Zeitschrift | volume = 285 | year = 2017 |issue = 3–4 | pages = 1041–1062| doi = 10.1007/s00209-016-1736-4 | arxiv = 1404.5522 | s2cid = 253752187 }}
• {{Citation| last=Steinberg | first = Robert | author-link = Robert Steinberg | title = Differential equations invariant under finite reflection groups | journal = Transactions of the American Mathematical Society | volume = 112 | pages = 392–400 | year = 1964 | issue = 3 | doi=10.1090/S0002-9947-1964-0167535-3}}
• {{citation | last = Taylor | first = D.E. | title = Reflection subgroups of finite complex reflection groups | journal = Journal of Algebra | volume = 366 | year = 2012 | pages = 218–234| doi = 10.1016/j.jalgebra.2012.04.033 | arxiv = 1201.1348 }}

Category:Coxeter groups

Category:Reflection groups