A Note on Element Centralizers in Finite Coxeter Groups
aa r X i v : . [ m a t h . G R ] S e p A NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETERGROUPS.
MATJAˇZ KONVALINKA, G ¨OTZ PFEIFFER, AND CLAAS E. R ¨OVER
Abstract.
The normalizer N W ( W J ) of a standard parabolic subgroup W J of afinite Coxeter group W splits over the parabolic subgroup with complement N J consisting of certain minimal length coset representatives of W J in W . In thisnote we show that (with the exception of a small number of cases arising froma situation in Coxeter groups of type D n ) the centralizer C W ( w ) of an element w ∈ W is in a similar way a semidirect product of the centralizer of w in a suit-able small parabolic subgroup W J with complement isomorphic to the normalizercomplement N J . Then we use this result to give a new short proof of Solomon’sCharacter Formula and discuss its connection to MacMahon master theorem. Introduction
Let W be a finite Coxeter group, generated by a set of simple reflections S withlength function ℓ : W → N ∪ { } . Each subset J ⊆ S generates a so-called standardparabolic subgroup W J = h J i of W . Conjugates of standard parabolic subgroupsare called parabolic subgroups. These subgroups are themselves Coxeter groups andtherefore play an important role in the structure theory of finite Coxeter groups.A well-known property of the cosets of a standard parabolic subgroup W J in W is that each coset contains a unique element of minimal length. The subgroup W J hence possesses a distinguished right transversal X J , consisting of the minimal lengthcoset representatives. Due to a theorem of Howlett [4] and later work of Brinkand Howlett [1], it is known that and how the normalizer N W ( W J ) of the parabolicsubgroup W J is a semidirect product of W J and a subgroup N J consisting of preciselythose minimal length coset representatives x ∈ X J which leave J as subset of W invariant in the conjugation action of W on its subsets, i.e., N J = { x ∈ X J : J x = J } .In this note we show that most centralizers of elements in W enjoy a similarsemidirect product decomposition. Pfeiffer and R¨ohrle [9] have shown, based onRichardson’s [10] characterization of involutions as central longest elements of par-abolic subgroups of W , that if w ∈ W is an involution then its centralizer in W coincides with the normalizer of a parabolic subgroup, and as such is a semidirect Key words and phrases.
Coxeter group, Solomon’s Character formula, MacMahon mastertheorem. product. This note can be regarded as a generalization of the result for involutions toall elements of W . Our results effectively reduce questions regarding the conjugacyclasses of elements in a finite Coxeter group W to the cuspidal conjugacy classes,that is those conjugacy classes which are disjoint from any proper parabolic sub-group of W . Cuspidal conjugacy classes of elements of W play a central role in thealgorithmic approach to the conjugacy classes of finite Coxeter groups in Chapter 3of the book by Geck and Pfeiffer [3]. We refer the reader to this book as a generalintroduction to the theory of finite Coxeter groups.We will call certain conjugacy classes of elements of a finite Coxeter group W non-compliant ; see Definition 4.6. Without exception, these are conjugacy classes of W which nontrivially intersect a parabolic subgroup of type D n with n >
4. Hence,if W has no parabolic subgroups of type D , part (ii) of the following theorem applieswithout restrictions. We can now formulate our main theorem as follows. Theorem 1.1.
Let W be a finite Coxeter group and let w ∈ W . Let V be the smallestparabolic subgroup of W that contains w . Then the following hold. (i) The centralizer C V ( w ) = C W ( w ) ∩ V is a normal subgroup of the centralizer C W ( w ) with quotient C W ( w ) /C V ( w ) isomorphic to the normalizer quotient N W ( V ) /V . (ii) The centralizer C W ( w ) splits over C V ( w ) with complement isomorphic to N W ( V ) /V unless w lies in a non-compliant conjugacy class of elementsof W . The parabolic subgroup V in the theorem is well-defined as the intersection ofall parabolic subgroups of W that contain w , due to the fact that intersections ofparabolic subgroups are parabolic subgroups, see Theorem 2.3 below. For the proofof the theorem, we will assume that w has minimal length in its conjugacy class in W . Then V is the standard parabolic subgroup W J of W , where J = J ( w ), the setof generators occurring in a reduced expression for w . The proof of part (i) is carriedout in Section 3. Part (ii) of the theorem is established case by case in Section 4. Theresults for the exceptional types of Coxeter groups have been obtained with the helpof computer programs using the GAP [11] package CHEVIE [2]. These programsare available through the second author’s ZigZag [8] package. In Section 5, we useTheorem 1.1 to provide a new short proof of a theorem of Solomon, and then discussits relation to MacMahon master theorem [5, page 98].2. Preliminaries.
In this section we recall some results about distinguished coset representatives andconjugacy classes in a finite Coxeter group W , generated by a set of simple reflections S and with length function ℓ . NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 3
For w ∈ W , we set J ( w ) = { s , . . . , s l } ⊆ S , if w = s . . . s l is a reduced expression,i.e., if l = ℓ ( w ). As a consequence of Matsumoto’s theorem, J ( w ) does not dependon the choice of a reduced expression for w .For w ∈ W , let D ( w ) = { s ∈ S : ℓ ( sw ) < ℓ ( w ) } be its descent set , and let A ( w ) = { s ∈ S : ℓ ( sw ) > ℓ ( w ) } = S \ D ( w )be its ascent set . The set X J = { w ∈ W : J ⊆ A ( w ) } is a right transversal for W J in W , consisting of the elements of minimal length ineach coset. For each element w ∈ W there are unique elements u ∈ W J and x ∈ X J such that w = u · x . Here the explicit multiplication dot indicates that the product ux is reduced , i.e., that ℓ ( ux ) = ℓ ( u ) + ℓ ( x ). An immediate consequence is the followinglemma. Lemma 2.1 ([3, Lemma 2.1.14]) . Let J ⊆ S . Then ℓ ( w x ) ≥ ℓ ( w ) for all w ∈ W J , x ∈ X J . We denote the longest element of W by w . For J ⊆ S , we denote by w J thelongest element of the parabolic subgroup W J . Lemma 2.2.
Let w ∈ W and let J = D ( w ) . Then w = w J · x for some x ∈ X J .Proof. This follows from [3, Lemma 1.5.2] and [3, Proposition 2.1.1] (cid:3)
For
J, K ⊆ S define X JK = X J ∩ X − K . Then X JK is a set of minimal lengthdouble coset representatives of W J and W K in W . Theorem 2.3 ([3, Theorem 2.1.12]) . Let
J, K ⊆ S and let x ∈ X JK . Then W xJ ∩ W K = W L , where L = J x ∩ K . Theorem 2.4 ([3, Theorem 2.3.3]) . Suppose
J, K are conjugate subsets of S andthat x ∈ X J is such that J x = K . If s ∈ D ( x ) then x = d · y , where d = w J w L for L = J ∪ { s } , and y ∈ X L . For the conjugacy classes of W , we are particularly interested in elements of min-imal length. These elements have useful properties, such as the following. Proposition 2.5 ([3, Corollary 3.1.11]) . Let C be a conjugcay class of W and let w, w ′ be elements of minimal length in C . Then J ( w ′ ) = J ( w ) x for some x ∈ X J ( w ) ,J ( w ′ ) . MATJAˇZ KONVALINKA, G ¨OTZ PFEIFFER, AND CLAAS E. R ¨OVER
A conjugacy class C of elements of W is called a cuspidal class if C ∩ W J = ∅ forall proper subsets J of S . Cuspidal classes never fuse in the following sense. Theorem 2.6 ([3, Theorem 3.2.11]) . Let J ⊆ S and let w ∈ W J be such that theconjugacy class C J of w in W J is cuspidal in W J . Then C J = C ∩ W J , where C is the conjugacy class of w in W . If w is of minimal length in its conjugacy class, then it is also of minimal lengthin its conjugacy class in the Coxeter group W J ( w ) , which by [3, Proposition 3.2.12] isa cuspidal class of W J ( w ) Below, we review some basic facts about Coxeter groups of classical type, thatis of type A , B or D . For a more detailed review of the combinatorics of the con-jugacy classes of finite Coxeter groups of classical type we refer the reader to thedescription [7] of the implementation of the character tables of these groups in GAP.2.1. Type A . Suppose W is a Coxeter group of type A n − . Then W is isomorphicto the symmetric group S n on the n points [ n ] = { , . . . , n } , with Coxeter generators s i = ( i, i +1), i = 1 , . . . , n −
1. The cycle type of a permutation w ∈ W is the partitionof n , which contains a part l for each l -cycle of w , where fixed points count as 1-cycles. Since any two elements of w are conjugate in W if and only if they have thesame cycle type, the conjugacy classes of elements of W are naturally parametrizedby the partitions of n .Here, it will be convenient to write partitions as weakly increasing sequences.Given a partition λ = ( λ , . . . , λ t ) of n (that is a sequence of positive integers λ ≤· · · ≤ λ t with λ + · · · + λ t = n ), there is a corresponding parabolic subgroup W J = S λ × · · · × S λ t containing an element w with cycle type λ . A particularelement of minimal length in this conjugacy class is the product w λ of t disjointcycles consisting of λ i successive points, for i = 1 , . . . , t . For example, a minimallength representative of the conjugacy class of elements with cycle structure 1124 in S is w = (1)(2)(3 , , , ,
8) = (3 , , , , w λ is a Coxeter element of W J , the product w λ = Y s i ∈ J s i (2.1)(in decreasing order) of all s i ∈ J . For example, J ( w ) = { s , s , s , s } , and w = s s s s .2.2. Type B . Suppose W is a Coxeter group of type B n . Then W is isomorphicto the group of permutations on {− n, . . . , − , , , . . . , n } satisfying w ( − i ) = − w ( i ).Alternatively, we can represent this group as the group of signed permutations, i.e. NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 5 injective maps from [ n ] to [ n ] ∪ − [ n ] with precisely one of i and − i in the im-age. Since the elements are permutations, we can write them in cyclic form. Wehave two types of cycles: cycles which do not contain i and − i for any i , and cy-cles in which i is an element if and only if − i is an element. Cycles of the firsttype come in natural pairs, and instead of ( i , i , . . . , i k )( − i , − i , . . . , − i k ), we write( i , i , . . . , i k ) and call it a positive cycle. Cycles of the second type are of the form( i , i . . . , i k , − i , − i , . . . , − i k ). We shorten that to ( i , i , . . . , i k ) − and call it a neg-ative cycle. For example, the permutation −
7→ − , − , − , − , ,
7→ − ,
7→ − ,
7→ − , , − , − − . In this notation, every signed permutation looks likean ordinary permutation in cyclic form, except that every element and every cyclecan have a minus sign. Note that we can change the sign of all elements in a cyclewithout changing the signed permutation. The Coxeter generators are t = (1) − and s i = ( i, i + 1), i = 1 , . . . , n −
1. We also set t i = ( i ) − , for i > w ∈ W ( B n ) can also be represented in the form of a signed permutationmatrix. This is an n × n matrix which acts on the standard basis { e , . . . , e n } of R n in the same way as the permutation w acts on the points [ n ] = { , . . . , n } , i.e., for i ∈ [ n ], it maps e i to e | w ( i ) | or its negative, depending on whether w ( i ) is positive ornegative. We will briefly use this matrix representation of W ( B n ) in Section 4.6.Since conjugation on a signed permutation in cyclic form works in the same wayas with usual permutations (if we conjugate with w , an element i of any cycle isreplaced by w ( i )), two signed permutations are conjugate if and only if they havethe same number of negative cycles of every length, and the same number of positivecycles of every length. The cycle type of a permutation w ∈ W is a double partition λ = ( λ + , λ − ) with | λ + | + | λ − | = n , so that λ + contains a part l for each positive l -cycle of w , and λ − contains a part l for each negative l -cycle of w . Two elementsof W are conjugate in W if and only if they have the same cycle type, and thereforethe conjugacy classes of elements of W are naturally parametrized by the doublepartitions of n . For example, the conjugacy class of (1 , , − , − (6 , − − is(21 , J ⊆ { t , s , s , . . . , s n − } and w ∈ W J . If s i / ∈ J , the elements i + 1 , . . . , n appear in positive cycles of w with all positive elements. Therefore, if we are givena double partition ( λ + , λ − ) of n , λ + = ( λ +1 , . . . , λ + t ), λ − = ( λ − , . . . , λ − s ), the smallestparabolic subgroup W J that contains an element of cycle type ( λ + , λ − ) is of the form W ( B | λ − | ) × S λ +1 × · · · × S λ + t . According to the description [3, 3.4.2] of conjugacyclasses of W , there is a minimal length representative w λ of the corresponding con-jugacy class of the following form. The negative cycles contain 1 , . . . , | λ − | , and the MATJAˇZ KONVALINKA, G ¨OTZ PFEIFFER, AND CLAAS E. R ¨OVER positive cycles contain | λ − | + 1 , . . . , n ; furthermore, each cycle contains only consecu-tive numbers in increasing order. For example, a minimal length representative of theconjugacy class corresponding to λ = (112 ,
23) is w λ = (1 , − (3 , , − (6)(7)(8 , Type D . Suppose W is a Coxeter group of type D n . Then W is isomorphic tothe group of permutations on {− n, . . . , − , , , . . . , n } satisfying w ( − i ) = − w ( i ) andwith an even number of i > w ( i ) <
0. Alternatively, we can represent thisgroup as the group of signed permutations with an even number of i mapping to − [ n ].These are precisely the signed permutations with an even number of negative cycles.The Coxeter generators are u = (1 , −
2) and s i = ( i, i + 1), i = 1 , . . . , n −
1. The cycletype of a permutation w ∈ W is a double partition ( λ + , λ − ) with | λ + | + | λ − | = n , sothat λ + contains a part l for each positive l -cycle of w , and λ − contains a part l foreach negative l -cycle of w . If two elements of W are conjugate, they have the samecycle type. Having the same cycle type, however, is not a sufficient condition forconjugacy. For example, u and s have the same cycle type but are not conjugate.It is easy to see that if they have the same cycle type ( λ + , λ − ) and | λ − | > λ + contains an odd part, they are conjugate. If they have the same cycle type ( λ + , ∅ ),where λ + contains only even parts, they are conjugate if and only if the number ofnegative numbers in their cycle decomposition has the same parity.We call a partition even if it consists of even parts only. The conjugacy classes ofelements of W are naturally parametrized by double partitions of n , where λ − hasan even number of parts, with two classes when λ − = ∅ and λ + is even. Given adouble partition ( λ + , λ − ) of n , λ + = ( λ +1 , . . . , λ + t ), λ − = ( λ − , . . . , λ − s ), s even, thecorresponding parabolic subgroup W J is of the form W ( D | λ − | ) × S λ +1 × · · · × S λ + t .Furthermore, there is a minimal length representative w λ of the corresponding conju-gacy class of the following form. The negative cycles contain 1 , . . . , | λ − | , the positivecycles contain | λ − | + 1 , . . . , n , and each cycle contains only consecutive numbers inincreasing order. If λ − = ∅ and λ + has only even parts, then there is an extra repre-sentative w ′ λ with the first positive cycle starting with − λ = (112 , w λ = (1 , − (3 , , − (6)(7)(8 , , ∅ ), we have w λ = (1 , , , , ,
8) and w ′ λ = ( − , , , , , Centralizers.
In this section we prove a general theorem about the structure of element central-izers in finite Coxeter groups. It is shown to be a consequence of Theorem 2.6, whichin the book [3] has been established by a careful case-by-case analysis. Without lossof generality, we may assume that w ∈ W is an element of minimal length in itsconjugacy class in W . NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 7
Theorem 3.1.
Suppose w ∈ W has minimal length in its conjugacy class in W andlet J = J ( w ) . Then C W ( w ) W J = N W ( W J ) . Clearly, part (i) of Theorem 1.1 follows from this result.
Proof.
Denote by C the conjugacy class of w in W and by C J its conjugacy classin W J , which is cuspidal in W J . By Theorem 2.6, C J = C ∩ W J , which impliesthat for every x ∈ N W ( W J ) there exists an element u ∈ W J with w x = w u . So xu − ∈ C W ( w ), i.e. x ∈ C W ( w ) W J , and hence N W ( W J ) ⊆ C W ( w ) W J .Now it only remains to show that C W ( w ) ⊆ N W ( W J ). Let y ∈ C W ( w ) andwrite it as y = uxv − for u, v ∈ W J and a double coset representative x ∈ X JJ .From w ∈ C J ∩ C yJ it then follows that w v ∈ C J ∩ C xJ ⊆ W J ∩ W xJ = W J ∩ J x , byTheorem 2.3, and since C J is a cuspidal class in W J , we must have J ∩ J x = J ,whence x ∈ N J ⊆ N W ( W J ) and thus y = uxv − ∈ N W ( W J ). (cid:3) The following additional results are of independent interest and will be used in theproof of Theorem 5.1 below.
Proposition 3.2.
Suppose w ∈ W has minimal length in its conjugacy class in W and let J = J ( w ) . Then the following hold. (i) The conjugacy class of w in W is a disjoint union of conjugates of the con-jugacy class of w in W J . (ii) If a ∈ C W ( w ) and x ∈ X J are such that C W J ( w ) a ⊆ W J x , then x ∈ N J . (iii) If x ∈ X J is such that ℓ ( w x ) = ℓ ( w ) , then J ( w x ) = J x . (iv) If v ∈ W is such that ℓ ( w v ) = ℓ ( w ) , then J ( w x ) = J x , where v = u · x with u ∈ W J and x ∈ X J .Proof. (i) and (ii) follow from the proof of Theorem 3.1.(iii) Let K = J ( w x ). By Proposition 2.5, there exists an element y ∈ X KJ suchthat K y = J . Hence w xy ∈ W J and ℓ ( w xy ) = ℓ ( w x ) = ℓ ( w ) and (since C ∩ W J = C J ) w xy = w u for some u ∈ W J . Moreover, X K = yX J . Hence u − xy centralizes w u , andif we write u − xy = a · z for a ∈ W J and z ∈ X J then, by (ii), z ∈ N J . It follows that zy − ∈ X JK is the unique minimal length representative of the coset W J x , hence x = zy − and J x = J ( w x ).(iv) We have w v = ( w u ) x . Conjugation with x does not decrease the length(Lemma 2.1), so ℓ ( w ) = ℓ ( w v ) ≥ ℓ ( w u ) and therefore ℓ ( w u ) = ℓ ( w ). By (iii), with w replaced by w u , we have J ( w v ) = J ( w u ) x , and J ( w u ) = J ( w ), which finishes theproof. (cid:3) MATJAˇZ KONVALINKA, G ¨OTZ PFEIFFER, AND CLAAS E. R ¨OVER Complements.
In this section we prove part (ii) of Theorem 1.1 for each type of irreducible finiteCoxeter group, case by case. We start with the general observation that part (ii) ofthe theorem is straightforward in the following situations.
Lemma 4.1.
Let w ∈ W be an element of minimal length in its conjugacy classin W , and let J = J ( w ) . If w is cuspidal in W or if C W J ( w ) = W J then N J is acomplement of C W J ( w ) in C W ( w ) .Proof. If w is cuspidal then W J = W and both quotients N W ( W J ) /W J and C W ( w ) /C W J ( w )are trivial.If C W J ( w ) = W J then w = w J and C W ( w ) = N W ( W J ) [9, Proposition 2.2]. (cid:3) Our general strategy in search of a centralizer complement for w will be to identifya complement M of W J in its normalizer that centralizes w . More precisely, we havethe following consequence of Theorem 3.1. Proposition 4.2.
Let w ∈ W be of minimal length in its conjugacy class, let J = J ( w ) and suppose that the normalizer complement N J is generated by elements x , . . . , x r . Let u , . . . , u r ∈ W J be such that u i x i ∈ C W ( w ) , i = 1 , . . . , r , and set M = h u x , . . . , u r x r i . Then M is a complement of C W J ( w ) in C W ( w ) provided that M ∩ W J = 1 .Proof. Clearly, W J M = W J N J = N W ( W J ). From M ∩ W J = 1 it then follows that M is a complement of W J in its normalizer. Moreover, M is a subgroup of C W ( w )since each of its generators centralizes w . From Theorem 3.1 it then follows that M is a complement of C W J ( w ) in C W ( w ). (cid:3) Type A . Let λ = (1 a , a , . . . , n a n ) be a partition of n , let w λ be as in (2.1)and let J = J ( w λ ). Then W J is a direct product W J = S a × S a × · · · × S a n n of symmetric groups, and its normalizer N W ( W J ) = S ≀ S a × S ≀ S a × · · · × S n ≀ S a n is a direct product of wreath products of symmetric with symmetric groups. In asimilar way, the centralizer C W ( w λ ) = C ≀ S a × C ≀ S a × · · · × C n ≀ S a n is a direct product of wreath products of cyclic with symmetric groups, and thecentralizer C W J ( w λ ) = C a × C a × · · · × C a n n . NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 9 is a direct product of cyclic groups. Clearly, the quotients N W ( W J ) /W J and C W ( w λ ) /C W J ( w λ )are both isomorphic to S a × S a × · · · × S a n . In order to show that the partic-ular normalizer complement N J is also a complement of C W J ( w λ ) in C W ( w λ ), weintroduce some notation. Let us define as s ( o, m ) = ( s o +1 s o +2 · · · s o +2 m − ) m = ( o + 1 , o + 2 , . . . , o + 2 m ) m (4.1)the permutation that swaps, after an offset o , two adjacent blocks of m points { o +1 , . . . , o + m } and { o + m + 1 , . . . , o + 2 m } . For example, s (2 ,
3) = ( s s s s s ) =(3 , , , , , = (3 , , , s i = s ( i − , N J = S a × S a × · · · × S a n is a direct product of symmetric groups S a m , with Coxeter generators s ( o m , m ), s ( o m + m, m ), . . . , s ( o m +( a m − m, m ), and offsets o m = a + 2 a + · · · + ( m − a m − , (4.2)for those m ∈ { , . . . , n } with a m > Proposition 4.3.
Let λ be a partition of n , let w λ be the permutation with cyclestructure λ from (2.1) and let J = J ( w λ ) be the corresponding subset of S . Then N J is a complement of C W J ( w λ ) in C W ( w λ ) .Proof. It suffices to consider the case λ = ( m a ) since all of W J , N W ( W J ), C W ( w λ ), C W J ( w λ ), and N J are subgroups of the direct product S a × S a × · · · × S na n inside S n and one can argue componentwise.If λ = ( m a ), then N J is isomorphic to S a , with a − s ( o, m ), s ( m, m ), . . . , s (( a − m, m ), permuting the blocks of m points { , . . . , m } , { m +1 , . . . , m } , . . . , { ( a − m +1 , . . . , am } .Clearly w λ = (1 , . . . , m )( m +1 , . . . , m ) · · · (( a − m +1 , . . . , am )is centralized by N J . The claim now follows from Theorem 3.1. (cid:3) Type B . Let λ be a double partition of n with λ + = (1 a , a , . . . , n a n ) and λ − = (1 b , b , . . . , n b n ), let w λ be as in Section 2.2, and let J = J ( w λ ) be thecorresponding subset of S . Then W J is a direct product W J = W ( B | λ − | ) × S a × S a × · · · × S a n n and its normalizer N W ( W J ) = W ( B | λ − | ) × S ≀ W ( B a ) × S ≀ W ( B a ) × · · · × S n ≀ W ( B a n ) is a direct product of W ( B | λ − | ) with wreath products of symmetric groups and Cox-eter groups of type B . In a similar way, the centralizer C W ( w λ ) = C W ( B | λ − | ) ( w λ ) × C ≀ W ( B a ) × C ≀ W ( B a ) × · · · × C n ≀ W ( B a n )is a direct product of C W ( B | λ − | ) ( w λ ) and wreath products, and the centralizer C W J ( w λ ) = C W ( B | λ − | ) ( w λ ) × C a × C a × · · · × C a n n is a direct product of C W ( B | λ − | ) ( w λ ) and cyclic groups. Clearly, the quotients N W ( W J ) /W J and C W ( w λ ) /C W J ( w λ ) are both isomorphic to W ( B a ) × W ( B a ) × · · · × W ( B a n ).In order to show that a variant of the particular normalizer complement N J is acomplement of C W J ( w λ ) in C W ( w λ ), we introduce some more notation.Denote by r ( o, m ) the permutation defined by x.r ( o, m ) = ( o + m + 1 − x, if o + 1 ≤ x ≤ o + m,x, otherwise.In this way, r ( o, m ) = ( o +1 , o + m )( o +2 , o + m − · · · reverses the range { o +1 , . . . , o + m } and thus is the longest element of the symmetric group S { o +1 ,...,o + m } with Coxetergenerators s o +1 , . . . , s o + m − . For example, r (2 ,
5) = (3 , , t ( o, m ) = ( o +1) − ( o +2) − · · · ( o + m ) − , which acts as − { o + 1 , o + 2 , . . . , o + m } and as identity everywhereelse.If λ + = ( m a ) and λ − = ∅ , then W J is a direct product of a copies of S m and N J is isomorphic to W ( B a ), with Coxeter generators r (0 , m ) t (0 , m ) and s (0 , m ), s ( m, m ), . . . , s (( a − m, m ).In general, if λ + = (1 a , a , . . . , n a n ), then W J is a direct product of W ( B | λ − | ) anddirect products of isomorphic symmetric groups S m and N J is a direct product ofgroups W ( B a m ), with Coxeter generators r ( o m , m ) t ( o m , m ) and s ( o m , m ), s ( o m + m, m ), . . . , s ( o m + ( a m − m, m )and offsets o m = (cid:12)(cid:12) λ − (cid:12)(cid:12) + a + 2 a + · · · + ( m − a m − , (4.3)for those m ∈ { , . . . , n } with a m >
0. Unfortunately, this group N J usually doesnot centralize w λ . However, if we define a group N λ as the subgroup of W generatedby the same elements as N J , with t ( o m , m ) in place of r ( o m , m ) t ( o m , m ), then N λ isa centralizing complement. NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 11
Proposition 4.4.
Let λ be a double partition of n , let w λ be as in Section 2.2, andlet J = J ( w λ ) be the corresponding subset of S . Then N λ = h t ( o m , m ) , s ( o m + km, m ) | k = 0 , . . . , a m − , m = 1 , . . . , n, a m > i is a complement of C W J ( w λ ) in C W ( w λ ) .Proof. Clearly, N λ centralizes w λ since its generators t ( o m , m ) and s ( o m + km, m )do. The statement now follows with Proposition 4.2 from the fact that N λ is acomplement of W J in its normalizer in W . (cid:3) Type D . The case of Coxeter groups of type D n is best dealt with by comparingit to the situation in type B n . Throughout this section, we assume n ≥
4, denoteby W the Coxeter group of type B n with Coxeter generators S , as described inSection 2.2, and by W + the Coxeter group of type D n with Coxeter generators S + ,consisting of the signed permutations with an even number of negative cycles asdescribed in Section 2.3.The following properties are easy to establish and we leave their proofs to thereader. Lemma 4.5.
Let w ∈ W be an element of cycle type λ = ( λ + , λ − ) such that λ − hasan even number of parts and that w has minimal length in its conjugacy class in W .Also let J = J ( w ) . Then the following hold. (i) w has minimal length in its class in W + . (ii) If λ − = ∅ and λ + is even then C W + ( w ) = C W ( w ) , otherwise C W + ( w ) hasindex in C W ( w ) . (iii) If J + = S + ∩ W J then W + J + is the smallest parabolic subgroup of W + con-taining w . (iv) If λ − = ∅ then J + = J , otherwise W + J + = W J ∩ W + is a subgroup of index in W J (v) C W + J + ( w ) = C W J ( w ) ∩ W + is a subgroup of index in C W J ( w ) unless λ − = ∅ . (vi) If λ + is not even and λ − = ∅ then N + J + = N J ∩ W + is a subgroup of index in N J , otherwise N J ∼ = N + J + . The parabolic subgroup W + J + is of the form D | λ − | × S λ +1 × · · · × S λ + t , where D m isthe subgroup of W + generated by { u, s , . . . , s m − } , for m = 2 , . . . , n . Definition 4.6.
We call a double partition λ = ( λ + , λ − ) a non-compliant doublepartition if λ + consists of a single odd part m and λ − is a nonempty even partitionof even length.We call a conjugacy class C of W a non-compliant class , if, for some odd n > λ of n and a parabolic subgroup W M of W which has an irreducible component W K of type D n , such that C contains anelement of W M whose projection on W K has cycle type λ .For example, the elements of W = W ( D ) with cycle type (1 ,
22) form a non-compliant class. For another example, the elements of W = W ( D ) of cycle type(21 ,
22) form a non-compliant class, as some of them lie in a parabolic subgroup W M of type D × A , with D -part of cycle type (1 , Lemma 4.7.
An element w ∈ W ( D n ) of cycle type λ = ( λ + , λ − ) lies in a non-compliant class if and only if λ + is not even and λ − is nonempty and even. The next result shows that, in a Coxeter group W + of type D n , the centralizer C W + ( w ) splits over C W + J ( w ), unless the class of w ∈ W + is non-compliant. Here,we write J + ( w ) ⊆ S + for the set of generators occurring in a reduced expressionof w when considered as an element of W + , in order to distinguish it from the set J ( w ) ⊆ S of generators in a reduced expression of w ∈ W . Proposition 4.8.
Let λ = ( λ + , λ − ) be a double partition of n be such that ℓ ( λ − ) iseven. Let w λ and N λ be as in Proposition 4.4 and let J + = J + ( w λ ) be the corre-sponding subset of S + . Then the following hold. (i) If λ + is even then N λ is a complement of C W + J + ( w λ ) in C W + ( w λ ) . (ii) If λ + is not even and λ − = ∅ then N λ ∩ W + is a subgroup of index in N λ and a complement of C W + J + ( w λ ) in C W + ( w λ ) . (iii) If λ + = (1 a , . . . , n a n ) and λ − = ( λ − , . . . , λ − s ) is not even then there is anindex j ≤ s such that k = λ − + · · · + λ − j is odd, and the subgroup N + λ = h t (0 , k ) m t ( o m , m ) , s ( o m + im, m ) | i = 0 , . . . , a m − , m = 1 , . . . , n, a m > i is a complement of C W + J + ( w λ ) in C W + ( w λ ) . Note that t (0 , k ) m = 1 if m is even and t (0 , k ) m = t (0 , k ) if m is odd. Proof.
Let J = J ( w λ ) be the subset of S corresponding to λ . In all three cases itsuffices to find a complement N ∗ of W J in its normalizer in W that centralizes w λ such that | N ∗ ∩ W + | = (cid:12)(cid:12) N + J + (cid:12)(cid:12) . For then N ∗ ∩ W + J + = 1 and N W + ( W + J + ) ⊆ N W ( W J ) = W J N ∗ imply that N ∗ ∩ W + is a complement of W + J + in its normalizer in W + thatcentralizes w λ , and the claim follows with Proposition 4.2.(i) If λ + is even then N λ is contained in W + and N ∗ = N λ will do.(ii) If λ − = ∅ and λ + is not even then J + = J but N J + is subgroup of index 2 in N J and N ∗ = N λ ∩ W + will do.(iii) If λ − is not even then N + λ is a complement of W J in its normalizer in W thatis contained in W + and centralizes w λ , whence N ∗ = N + λ will do. (cid:3) NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 13
Type I . Suppose W is a Coxeter group of type I ( m ). Then W is the groupgenerated by generators s and s satisfying ( s s ) m = ( s s ) m . Each element of W is either cuspidal or an involution. Hence the theorem for this type follows fromLemma 4.1.4.5. Exceptional Types.
Although in type A each conjugacy class contains anelement w such that the normalizer complement N J is also a centralizer complement,this cannot be expected in general to be the case. However, from the precedingexamples one sees that it is frequently possible to construct from N J an isomorphiccopy N ∗ J which is a centralizer complement. In each of the above examples, N ∗ J isobtained from N J by replacing generators x i of N J by products w L x i for suitablesubsets L ⊆ J .Based on this observation, we formulate an algorithm, which in practice alwaysfinds a centralizer complement, except for elements of non-compliant classes. Algorithm
CentralizerComplement.
Input:
A finite Coxeter group W and an element w of minimal length in its conju-gacy class in W . Output: a centralizer complement for w , or fail if none exists.1. set J ← J ( w ).2. find involutions x , . . . , x r generating the normalizer complement N J .3. for each element v of minimal length in the W J -conjugacy class of w do thefollowing: • let u ∈ W J be such that v u = w ; • for each i = 1 , . . . , r , set Y i ← ( { x i } , if v x i = v, { w L x i : L ⊆ J, x w L i = x i , v w L = v x i } , otherwise. • if there are elements y i ∈ Y i , i = 1 , . . . , r , such that M = h y , . . . , y r i satisfies M ∩ W J = 1 then return M u .4. return fail (if we ever get here).Note that, by Proposition 4.2, any group M found in this way is necessarily acomplement of the centralizer of w in W J .For W irreducible of exceptional type, the algorithm produces a centralizer com-plement in all but seven cases. Each case corresponds to a non-compliant class fromthe following table. In this table we list, for each non-compliant class C of W , itsposition i in CHEVIE’s list of conjugacy classes of W , its name, a reduced expressionfor a representative w of minimal length, the set J ( w ), a set M ⊇ J ( w ), the type of W M exhibiting a direct factor of type D l +1 , and the label λ of the conjugacy classof W ( D l +1 ) containing the projection of w . W i name w ∈ C J ( w ) M type λE D ( a ) D (1 , E D ( a ) D (1 , D ( a ) + A D × A (1 , E D ( a ) D (1 , D ( a ) D (1 , D ( a ) + A D × A (1 , D ( a ) + A D × A (1 , w liesin a non-compliant class, no complement exists.4.6. Non-compliance.
The class of W ( D ) with label λ = (1 ,
22) contains theelement w λ = ts s s ts s s = ( ts t ) s ( ts t ) s s s = us us s s , which lies in the parabolic subgroup W J of type D . Its centralizer C W J ( w ) in W ( D )has order 16 and its centralizer C W ( w ) in W ( D ) has order 32. However, C W J ( w )has no complement in C W ( w ), since the coset C W ( w ) \ C W J ( w ) contains no elementof order 2 (as a straightforward computation in GAP will confirm).The next result shows that this is indeed always the case, when the cycle type of w ∈ W ( D n ) is a non-compliant double partition of n . Proposition 4.9.
Suppose that W is of type D n and let w ∈ W be an element ofminimal length in its conjugacy class with J ( w ) = J . If the cycle type of w ∈ W is anon-compliant double partition of n then the centralizer C W J ( w ) has no complementin C W ( w ) .Proof. Recall from Section 2.2 that elements of W ( B n ) can be represented as signedpermutation matrices, i.e., matrices with exactly one non-zero entry 1 or − W ( D n ) if and only if its matrix has an evennumber of entries −
1, and it is an involution if and only if the matrix is symmetric.Now suppose that n = m + k is odd and that w ∈ W = W ( D n ) is an element ofminimal length in a conjugacy class with cycle type λ = ( λ + , λ − ) where the partition λ + consists of a single odd part m and λ − is a nontrivial partition of an even number k , consisting of an even number of even parts. Then W J for J = J ( w ) has type D k × A m − and, by the description of N J in Section 4.2 and Lemma 4.5(vi), itsnormalizer N W ( W J ) has a complement of order 2 (and of type B ), generated by thequotient w J w . NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 15
We may assume that J = S \ { s k +1 } , so that, as signed permutation on the set { , . . . , n } , the element w induces an even number of negative cycles on the k points { , . . . , k } and a positive m -cycle on the m points { k + 1 , . . . , n } . The centralizer C W ( w ) cannot move points from outside the m -cycle into the m -cycle and thusconsists of block diagonal matricesdiag( A, B ) = (cid:20) A B (cid:21) , of a k × k matrix A and an m × m -matrix B , which modulo 2 have the same numberof entries − C W ( w ) is a subgroup of W ( D n ). Moreover, for each elementdiag( A, B ) in C W ( w ) the number of entries − A , is even, sincewith every point in { , . . . , k } being mapped to its negative, the entire cycle whichcontains it must be negated.The centralizer of w in W J consists precisely of those elements diag( A, B ) ∈ C W ( w )which have an even number of entries − A and B , since A is the matrix ofan element in W ( D k ).Let u be an involution in C W ( w ). Then its matrix diag( A, B ) is symmetric, andan even number of entries − A implies that both A and B havean even number of entries − u ∈ C W J ( w ).It follows that C W J ( w ) has no complement in C W ( w ). (cid:3) More generally, if w lies in a non-compliant class of a finite Coxeter group W , thenits centralizer has no complement. Theorem 4.10.
Let W be a finite Coxeter group. Suppose w is an element ofminimal length in a non-compliant conjugacy class of W with J ( w ) = J . Then thecentralizer C W J ( w ) has no complement in C W ( w ) .Proof. Suppose first that W is a direct product W × W of nontrivial standardparabolic subgroups W and W , that w = w w with w ∈ W and w ∈ W , andthat w lies in a non-compliant class of W . Then W J = W J × W J for certainsubsets J ⊆ W ∩ S and J ⊆ W ∩ S . If C W J ( w ) has no complement in C W ( w )then C W J ( w ) cannot have a complement in C W ( w ) = C W ( w ) × C ( w ).Next, suppose that w ∈ W L for some L ⊆ S and that w lies in a non-compliantclass of W L . Suppose N is a complement of C W J ( w ) in C W ( w ), that is C W ( w ) = C W J ( w ) ⋊ N . Then the centralizer of w in W L , C W L ( w ) = C W ( w ) ∩ W L = ( C W J ( w ) ⋊ N ) ∩ W L = C W J ( w ) ⋊ ( N ∩ W L ) , in contradiction to our assumption that w lies in a non-compliant class.The theorem now follows from Definition 4.6 and Proposition 4.9. (cid:3) Applications.
In this section we first use Theorem 1.1 to prove a result about minimal lengthrepresentatives of conjugacy classes. Then we show how it implies the celebratedSolomon’s character formula. Finally, we discuss the interpretation of Solomon’stheorem as a Coxeter group analogue of MacMahon master theorem.
Theorem 5.1.
Assume w has minimal length in its conjugacy class in W . Then thefollowing hold for any v ∈ W : (i) J ( w v ) = D ( v − ) ⇐⇒ v = w J ( w ) ; (ii) J ( w v ) = A ( v − ) ⇐⇒ v = w J ( w ) w .Proof. Let L = D ( v − ). Then v − = w L · x for some x ∈ X L , by Lemma 2.2. Clearly ℓ (( w v ) w L ) = ℓ ( w v ), since J ( w v ) = L . By Lemma 2.1, conjugation by the coset rep-resentative x does not decrease the length, hence ℓ ( w v ) = ℓ (( w v ) w L ) ≤ ℓ (( w v ) w L x ) = ℓ ( w ) and it follows that w v has minimal length in its conjugacy class as well. ByProposition 2.5, L = J ( w v ) and J ( w ) are conjugate subsets of S . Proposition 3.2(iv) says more precisely that x is a conjugating element, i.e., L x = J ( w ).Assume that ℓ ( x ) > s ∈ D ( x ). Then s / ∈ L , since x ∈ X L ; denote L ∪ { s } by M . By Theorem 2.4, x is a reduced product x = d · y with y ∈ X M and d = w L w M ,the longest coset representative of W L in W M . It follows that v − = w L · x = w M · y ,whence M ⊆ D ( v − ) = L ( M . The contradiction shows that x = 1, and therefore v = w L and L = J ( w ).(ii) Note that J ( w w ) = J ( w ) w , D ( x w ) = D ( x ) w , and A ( x ) = D ( xw ) for all x ∈ W , whence D ( x ) w = A ( w x ). Therefore, it follows from (i) that J ( w xw ) = J ( w x ) w = D ( x − ) w = A (( xw ) − ) ⇐⇒ x = w J ( w ) , as desired, for v = xw . (cid:3) The following formula, first proved by Solomon [12] in 1966, is an easy consequenceof the previous result.
Theorem 5.2 (Solomon’s theorem) . For J ⊆ S , let π J denote the permutationcharacter of the action of W on the cosets of W J defined by π J ( w ) = (cid:12)(cid:12) Fix
W/W J ( w ) (cid:12)(cid:12) ,and let ǫ be the sign character of W , defined by ǫ ( w ) = ( − ℓ ( w ) for w ∈ W . Then X J ⊆ S ( − | J | π J = ǫ. NOTE ON ELEMENT CENTRALIZERS IN FINITE COXETER GROUPS. 17
Proof.
The formula follows if we can show that P J ⊆ S ( − | J | π J ( w ) = ǫ ( w ) for all w ∈ W . We have W J xw = W J x ⇐⇒ xwx − ∈ W J , so X J ⊆ S ( − | J | π J ( w ) = X J ⊆ S ( − | J | X x ∈ X J xwx − ∈ W J X x ∈ W X J ( xwx − ) ⊆ J ⊆A ( x ) ( − | J | , where we reversed the order of summation and used the facts that x ∈ X J ⇐⇒ J ⊆A ( x ) and xwx − ∈ W J ⇐⇒ J ( xwx − ) ⊆ J . The Binomial Theorem implies that X A ⊆ J ⊆ B ( − | J | = ( − | A | X I ⊆ B \ A ( − | I | = ( ( − | A | if A = B, X J ⊆ S ( − | J | π J ( w ) = X x ∈ WJ ( xwx − )= A ( x ) ( − |A ( x ) | . Since P J ⊆ S ( − | J | π J is a class function, it is enough to choose w with minimal lengthin its conjugacy class. By Theorem 5.1(ii), this sum then consists only of the oneterm for x − = w J ( w ) w , and we have X J ⊆ S ( − | J | π J ( w ) = ( − | A ( w w J ( w ) ) | . But (cid:12)(cid:12) A ( w w J ( w ) ) (cid:12)(cid:12) = (cid:12)(cid:12) D ( w J ( w ) ) w (cid:12)(cid:12) = (cid:12)(cid:12) D ( w J ( w ) ) (cid:12)(cid:12) = | J ( w ) | and then the claim followsfrom the fact that ( − | J ( w ) | = ( − ℓ ( w ) [3, Exercise 3.17]. (cid:3) Solomon proves this formula generically for all types of finite Coxeter groups, andhe has published three different versions of the proof. His original proof [12] dependson an application the Hopf trace formula to the Coxeter complex of the finite Coxetergroup W , a later proof (of a more general statement) uses a decomposition of thegroup algebra of W . The third version of the proof [14] is based on properties of ahomomorphism of the descent algebra of W into the character ring of W (see also[3, Exercise 3.15]). None of these proofs have the combinatorial flavor of the aboveproof.Finally, let us explain how to interpret Solomon’s theorem as a generalization ofa special case of the celebrated MacMahon master theorem. For a connection witha different result due to MacMahon, see [13, § X = ( x ij ) n × n , the functions(5.1) 1det(Id − X ) , where Id denotes the n × n -identity matrix, and X w x v w x v w · · · x v m w m , where w = w w · · · w m runs over all words in { , , . . . , n } and v = v v · · · v m is theweakly increasing rearrangement of w , are equal. In particular, for any permutation w ∈ S n , the coefficient of x w (1) · · · x nw ( n ) in (5.1) is equal to 1. For I ⊆ [ n ], denoteby X I the submatrix ( x ij ) i,j ∈ I . We have1det(Id − X ) = 1 P I ⊆ [ n ] ( − | I | det X I = 11 − P ∅ = I ⊆ [ n ] ( − | I |− det X I = X k ≥ (cid:16) X ∅ = I ⊆ [ n ] ( − | I |− det X I (cid:17) k = X k ≥ X ( − | I |− ... + | I k |− det X I · · · det X I k , where the last sum runs over all k -tuples ( I , . . . , I k ) of non-empty subsets of [ n ].Since we are interested in the coefficient of x w (1) · · · x nw ( n ) (in which all indices arerepresented, and each index is represented only twice, once as a first index and onceas a second index), we can limit the sum to ordered set partitions ( I , . . . , I k ) of theset [ n ]. Note that we have ( − | I |− ... + | I k |− = ( − n − k .Recall that the symmetric group S n is a Coxeter group W of type A n − withCoxeter generators S = { s , . . . , s n − } , s i = ( i, i + 1). Choose a composition λ ⊢ n .By Merris-Watkins formula [6] (and not hard to prove independently), the coefficientof x w (1) · · · x nw ( n ) in X det X I · · · det X I k , where the sum runs over all ordered set partitions ( I , . . . , I k ) of [ n ] with | I j | = λ j for all j , is equal to ( − ℓ ( w ) π J ( w ), where J is the subset of S that corresponds tothe composition λ . This means that X ( − n −| λ | ( − ℓ ( w ) π J ( w ) = 1 , where the sum runs over all subsets J of S . This is obviously equivalent to Solomon’stheorem. Acknowledgement.
The second author wishes to acknowledge support from Sci-ence Foundation Ireland.
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