Bounding reflection length in an affine Coxeter group
aa r X i v : . [ m a t h . C O ] O c t BOUNDING REFLECTION LENGTH INAN AFFINE COXETER GROUP
JON MCCAMMOND AND T. KYLE PETERSEN
Abstract.
In any Coxeter group, the conjugates of elements inthe standard minimal generating set are called reflections and theminimal number of reflections needed to factor a particular elementis called its reflection length. In this article we prove that thereflection length function on an affine Coxeter group has a uniformupper bound. More precisely we prove that the reflection lengthfunction on an affine Coxeter group that naturally acts faithfullyand cocompactly on R n is bounded above by 2 n and we also showthat this bound is optimal. Conjecturally, spherical and affineCoxeter groups are the only Coxeter groups with a uniform boundon reflection length. Every Coxeter group W has two natural generating sets: the finite set S used in its standard finite presentation and the set R of reflectionsformed by collecting all conjugates of the elements in S . The firstgenerating set leads to the standard length function ℓ S : W → N andthe second is used to define the reflection length function ℓ R : W → N .When W is finite, both length functions are uniformly bounded fortrivial reasons and are fairly well understood. For infinite Coxetergroups the function ℓ S is always unbounded because there are onlyfinitely many group elements of a given length as a consequence of thefact that S is finite. Our main result is that ℓ R remains bounded foraffine Coxeter groups and we provide an explicit optimal upper bound. Theorem A (Explicit affine upper bounds) . If W is an affine Coxetergroup that naturally acts faithfully and cocompactly on R n then everyelement of W has reflection length at most n and there exist elementsin W with reflection length equal to n . Date : November 11, 2018.Work of McCammond partially supported by an NSF grant.Work of Petersen partially supported by an NSA Young Investigator grant. For finite W these generating sets and length functions exhibit an interestingduality: the maximum value of ℓ S is | R | and the maximum value of ℓ R is | S | . See[1] for further details and for additional illustrations of this phenomenon. The article is structured as follows. The first two sections recallbasic facts, the third establishes a key technical result and the fourthcontains the proof of our main result. The final section contains aconjecture about infinite non-affine Coxeter groups.1.
Reflection length
We assume the reader is familiar with the basic theory of reflectiongroups (as described, for example, in [4]) and we generally follow thestandard notational conventions.
Definition 1.1 (Reflection length) . Let W be a Coxeter group withstandard generating set S . A reflection in W is any conjugate of anelement of S and we use R to denote the set of all reflections in W .In other words, R = { wsw − | s ∈ S, w ∈ W } . We should note thatunless W is a finite group, R is an infinite set. For any element w ∈ W ,its reflection length ℓ R ( w ) is the minimal number of reflections whoseproduct is w . Thus w = r r · · · r k with r i ∈ R means ℓ R ( w ) ≤ k .Alternatively, ℓ R ( w ) can be defined as the combinatorial distance fromthe vertex labeled by the identity to the vertex labeled by w in theCayley graph of W with respect to R .Since combinatorial distance defines a metric on the vertex set ofany graph and Cayley graphs are homogeneous in the sense that thereis a vertex-transitive group action, metric properties of the distancefunction translate into properties of ℓ R . Symmetry and the triangleinequality, for example, imply that ℓ R ( w ) = ℓ R ( w − ), and ℓ R ( uv ) ≤ ℓ R ( u ) + ℓ R ( v ), respectively.It is sufficient to investigate reflection length in irreducible Coxetergroups because of the following elementary result. Proposition 1.2 (Reducible Coxeter groups) . When W is a reducibleCoxeter group, its standard generating set S has a nontrivial partition S = S ⊔ S in which every element in S commutes with every elementin S . In this context, W = W × W where W i denotes the parabolicsubgroup generated by S i , the reflections R in W can be partitionedas R = R ⊔ R where R i denotes the reflections in W i and when w ∈ W is written in the form w = w w with w i ∈ W i , we have ℓ R ( w ) = ℓ R ( w ) + ℓ R ( w ) . Affine Coxeter groups
Next we review the construction of an affine Coxeter group from acrystallographic root system.
EFLECTION LENGTH 3
Definition 2.1 (Affine Coxeter groups) . Recall that a crystallographicroot system Φ is a finite collection of vectors that span a real Euclideanspace V satisfying a few elementary properties and that an affine Cox-eter group W can be constructed from Φ as follows. For each α ∈ Φand i ∈ Z let H α,i denote the (affine) hyperplane of solutions to theequation h x, α i = i where the brackets denote the standard inner prod-uct on V . The unique nontrivial isometry of V that fixes H α,i pointwiseis a reflection that we call r α,i . The collection R = { r α,i | α ∈ Φ , i ∈ Z } generates the affine Coxeter group W and R is its set of reflections inthe sense of Definition 1.1. A standard minimal generating set S canbe obtained by restricting to those reflections that reflect across thefacets of a certain polytope in V . Remark 2.2 (No finite factors) . Every irreducible affine Coxeter groupcan be constructed from its crystallographic root system as describedin Definition 2.1 but the construction also works equally well whenthe root system is reducible. It is not, however, sufficient to constructarbitrary reducible affine Coxeter groups because it always constructsaffine Coxeter groups with affine irreducible components (i.e. no fi-nite irreducible components). The affine Coxeter groups constructiblefrom a root system in this way can also be characterized as those thatnaturally act faithfully and cocompactly on some Euclidean space.For each affine Coxeter group W constructed from a root system Φthere is a finite Coxeter group W related to W in two distinct ways. Definition 2.3 (Subgroups and quotients) . We abbreviate r α, and H α, as r α and H α , respectively. The hyperplanes H α are precisely theones that contain the origin and the reflections R = { r α | α ∈ Φ } generate a finite Coxeter group W that contains all elements of W that fix the origin. This embeds W as a subgroup of W . There is agroup homomorphism p : W ։ W defined by sending each generatingreflection r α,i in W to r α in W .Because the map p : W ։ W sends R to R , it sends reflectionfactorizations to reflection factorizations, proving the following. Proposition 2.4 (Lengths and quotients) . If the map p : W ։ W sends u to u then ℓ R ( u ) ≥ ℓ R ( u ) . Finite Coxeter groups such as W are also known as spherical Cox-eter groups because they act by isometries on spheres and we digressfor a moment to recall a few of their key properties. In a finite Coxetergroup, reflection length has a geometric interpretation that yields aspherical version of Theorem A as an immediate corollary. The follow-ing properties were observed by Carter in [2]. J. MCCAMMOND AND T. K. PETERSEN
Proposition 2.5 (Spherical reflection length) . The reflection length ofan element w in a finite Coxeter group W is equal to the codimensionof the subspace of vectors that w fixes in the standard orthogonal repre-sentation of W . In addition, a reflection factorization w = r r · · · r m is of minimum length if and only if the vectors normal to the hyper-planes of these reflections are linearly independent. Corollary 2.6 (Spherical upper bounds) . Let W be a finite Coxetergroup whose standard representation acts on R n by orthogonal trans-formations. For all w ∈ W , ℓ R ( w ) ≤ n and for elements that onlyfix the origin, ℓ R ( w ) = n . More concretely, multiplying the elementsin its standard minimal generating set S produces an element w with ℓ R ( w ) = n . Proposition 2.5 also has consequences for some elements in W . Corollary 2.7 (Linearly independent roots) . If w = r r · · · r m is areflection factorization of w ∈ W in which the roots of the reflectionsare linearly independent, then ℓ R ( w ) = m .Proof. The factorization shows ℓ R ( w ) ≤ m . For the lower bound notethat each reflection r i ∈ R is r α i ,c i for some α i ∈ Φ and c i ∈ Z , and byhypothesis the roots α i are linearly independent. If we let w = p ( w )then by Propositions 2.4 and 2.5 we have ℓ R ( w ) ≥ ℓ R ( w ) = m . (cid:3) Although we never use the following result established by Solomon in[6], we mention it because it highlights how reflection length capturesfundamental aspects of the behavior of finite Coxeter groups.
Proposition 2.8 (Solomon’s factorization formula) . For each finiteCoxeter group W , the polynomial recording the distribution of reflec-tion lengths factors completely over the integers. In particular, f ( x ) = X w ∈ W x ℓ R ( w ) = n Y i =1 (1 + e i x ) , where the e i are the exponents of W . We now return to the structure of the affine Coxeter group W con-structed from a root system Φ. Definition 2.9 (Coroots) . For each root α ∈ Φ, there is a correspond-ing coroot α ∨ = cα with c = h α,α i . The collection of all coroots isdenoted Φ ∨ and the integral linear combinations of vectors in Φ ∨ isa lattice L = Z Φ ∨ ∼ = Z n called the coroot lattice . The isomorphismwith Z n is a result of the existence of a subset ∆ ∨ ⊂ Φ ∨ of linearlyindependent vectors that form a basis for L . EFLECTION LENGTH 5
The coroot lattice describes the translations in W . Definition 2.10 (Translations) . A translation is a map that shiftseach point by the same vector λ and we let t λ denote the map sendingeach point x ∈ V to x + λ . For each α ∈ Φ consider the product r α, r α . Reflecting through parallel hyperplanes produces a translationin the α direction and the exact translation is t α ∨ . Because t µ t ν = t µ + ν ,there is a translation of the form t λ in W for each λ ∈ L and the set T = { t λ | λ ∈ L } forms an abelian subgroup of W . In fact, these arethe only translations that are contained in W .The subgroup T is also the kernel of the map p : W ։ W and W can be viewed as a semidirect product W = W ⋉ T . Proposition 2.11 (Normal forms) . For each element w ∈ W there isa unique factorization w = t λ w where t λ is a translation with λ ∈ L and w is an element in W .Proof. If such an expression exists then w is the image of w under themap p : W ։ W and λ is the image of the origin under w (keeping inmind that elements of W act on V in function notation so that com-position is from right to left). This proves uniqueness. For existencedefine w = p ( w ) ∈ W ⊂ W and consider the element ww − . Since itis in the kernel of p , it is translation in the form t λ for some λ ∈ L . (cid:3) Translation dimension
In this section we introduce the notion of the dimension of a vector inthe coroot lattice. As above, W is an affine Coxeter group, constructedfrom a root system Φ, acting on a vector space V . Definition 3.1 (Real dimension) . We call a subspace of V spanned bya collection of coroots in Φ ∨ a real coroot subspace and we say a vector λ has real dimension k when λ is contained in a k -dimensional corootsubspace but it is not contained in any real coroot subspace of strictlysmaller dimension. Since it is always possible find a coroot basis foreach coroot subspace, real dimension k means that λ = c α ∨ + c α ∨ + · · · + c k α ∨ k for some c i ∈ R and α ∨ i ∈ Φ ∨ but that no such expressionexists with fewer summands. Remark 3.2 (Non-unique subspaces) . There may be more than one k -dimensional coroot subspace containing a vector λ ∈ L of real di-mension k because coroot subspaces are not closed under intersection.Consider the D n root system Φ = {± e i ± e j | ≤ i < j ≤ n } .Every root has norm 2 so α = α ∨ and Φ = Φ ∨ . The translation t λ with λ = 2 e ∈ L is 2-dimensional because 2 e is in the span of J. MCCAMMOND AND T. K. PETERSEN { e + e j , e − e j } for any j > j varies we get distinct 2-planescontaining λ . They intersect along the e -axis but this line is not a realcoroot subspace.The following definition is very similar. Definition 3.3 (Integral dimension) . We say a vector λ ∈ L has in-tegral dimension k when λ can be expressed in the form λ = c α ∨ + c α ∨ + · · · + c k α ∨ k for some c i ∈ Z and α ∨ i ∈ Φ ∨ but no such expressionexists with fewer summands. The restriction to integral coefficientsmeans that only vectors in L have a dimension in this sense. When λ has integral dimension k we say λ is k -dimensional as is the corre-sponding translation t λ . Note that the real dimension of λ is a lowerbound on its integral dimension. Proposition 3.4 (Dimensions) . If W is an affine Coxeter group con-structed from a root system Φ and naturally acts faithfully and cocom-pactly on V = R n , then every vector in its coroot lattice has integraldimension at most n and n -dimensional vectors do exist.Proof. The first assertion is a consequence of the fact that L ∼ = Z n isa lattice with a Z -basis in Φ ∨ . For the second assertion note that anyvector λ ∈ L that does not lie in the union of the finite number ofproper subspaces through the origin that are spanned by coroots hasreal dimension n and thus integral dimension at least n . (cid:3) The integral dimension of a vector λ ∈ L bounds how hard it is tomove the origin to λ using reflections. To prove this assertion we needan elementary result about factorizations. Lemma 3.5 (Rewriting factorizations) . Let W be a Coxeter groupwith reflections R and let w = r r · · · r m be a reflection factorization.For any selection ≤ i < i < · · · < i k ≤ m of positions there isa length m reflection factorization of w whose first k reflections are r i r i · · · r i k and another length m reflection factorization of w wherethese are the last k reflections in the factorization.Proof. Because reflections are closed under conjugation, for any reflec-tions r and r ′ there exist reflections r ′′ and r ′′′ such that rr ′ = r ′′ r and r ′ r = rr ′′′ . Iterating these rewriting operations allows us to movethe selected reflections into the desired positions without altering thelength of the factorization. (cid:3) In preparation for the next result recall that t α ∨ = r α, r α and notethat because the hyperplanes H α,i are evenly spaced, the product r α,i +1 r α,i is also t α ∨ for every i ∈ Z . More generally r α,i + j r α,i = t jα ∨ . EFLECTION LENGTH 7
Proposition 3.6 (Moving points) . If λ ∈ L has integral dimension k then ℓ R ( t λ ) ≤ k and there is an element u ∈ W with ℓ R ( u ) ≤ k thatsends the origin to λ .Proof. By the definition of integral dimension there is an equation ofthe form λ = c α ∨ + c α ∨ + · · · + c k α ∨ k and the formulas t µ + ν = t µ t ν and t jα ∨ = r α,j r α show that t λ has a length 2 k reflection factorization ofthe form t λ = ( r α ,c r α )( r α ,c r α ) · · · ( r α k ,c k r α k ). Next, by Lemma 3.5there is another length 2 k reflection factorization of t λ where the final k reflections are r α r α · · · r α k . Since all of these reflections fix the origin,the product of the first k reflections is an element u that sends theorigin to λ . (cid:3) Note that just as there can be distinct minimal expressions for λ (Remark 3.2), there are often distinct elements of reflection length k that send the origin to λ . Theorem 3.7 (Equivalent definitions) . For each λ ∈ L the real di-mension of λ , the integral dimension of λ and the minimal reflectionlength of an element sending the origin to λ are equal.Proof. Let k r , k d and k m be three numbers at issue in the order listed.Proposition 3.6 shows that k d ≥ k m . Next, let u be an element sendingthe origin to λ . Because a reflection r α,i move points in the α ∨ direction, λ is in the span of the coroots associated to the reflections in a minimallength reflection factorization of u . Thus k m ≥ k r . And finally, let V ′ be a k r -dimensional coroot subspace of V containing λ . The set Φ ′ =Φ ∩ V ′ satisfies the requirements to be a root system and L ′ = L ∩ V ′ is its coroot lattice. Because λ lies in the lattice L ′ , k r ≥ k d . Thecombination k d ≥ k m ≥ k r ≥ k d shows all three are equal. (cid:3) Bounding reflection length
We are now ready to prove our main result.
Proposition 4.1 (Bounds) . If W is a affine Coxeter group and w ∈ W has the form w = t λ w where t λ is a k -dimensional translation and w ∈ W is an element fixing the origin then k ≤ ℓ R ( w ) ≤ k + n . Inparticular, every element w ∈ W has ℓ R ( w ) ≤ n .Proof. The lower bound, ℓ R ( w ) ≥ k , follows from Theorem 3.7. ByProposition 3.6 and Theorem 3.7 there is an element u with ℓ R ( u ) = k that sends the origin to λ . Because the element v = u − w fixes theorigin, it is in W and ℓ R ( v ) ≤ n by Corollary 2.6. Thus ℓ R ( w ) = ℓ R ( uv ) ≤ ℓ R ( u ) + ℓ R ( v ) ≤ k + n . For the final assertion note that J. MCCAMMOND AND T. K. PETERSEN every element can be written in this form (Proposition 2.11) and everytranslation is k -dimensional for some k ≤ n (Proposition 3.4). (cid:3) Based on the proof of this proposition, one might conjecture thateach w = t λ w with k -dimensional λ has another factorization w = uv with v ∈ W and ℓ R ( w ) = ℓ R ( u ) + ℓ R ( v ). This is not always the case. Example 4.2 (Exact bounds) . Let α ij denote the vector e i − e j , letΦ = { α ij | ≤ i, j ≤ } be the A root system with coroot lattice L ,and consider the elements w ijk = r α ,i r α ,j r α ,k . By Corollary 2.7 ℓ R ( w ijk ) = 3. All of these elements are sent under p : W ։ W to w = r α r α r α which cyclically permutes the four coordinates mov-ing the point ( x, y, z, w ) to ( w, x, y, z ). The minimum length reflectionfactorizations of w in W are well-known and they are encoded bymaximal chains in the lattice N C of non-crossing partitions on fourelements [5]. Every reflection in R occurs in some factorization of w but there exists a pair of reflections, r α and r α , representing a“crossing” partition that cannot both occur in the same factorization.By varying i , j and k , the elements w ijk produce all elements of theform t λ w with λ ∈ L . Thus for a careful choice of values, the vector λ is in the span of α and α and this is the unique minimal dimen-sional coroot subspace containing λ . If u is a product of two reflectionssending such a λ to the origin, then the coroots involved in this productmust be α and α . As a consequence, there is no length 3 reflectionfactorization of the form w ijk = uv since the projection of such a fac-torization to W would be a length 3 reflection factorization of w containing both r α and r α which is known not to exist. Proposition 4.3 (Optimality) . If W is a affine Coxeter group and w = t λ is a k -dimensional translation then ℓ R ( w ) = 2 k . In particular, n -dimensional translations have reflection length n .Proof. Being a translation in W , w = t λ for some λ ∈ L and by defini-tion of dimension λ = c α ∨ + c α ∨ + · · · + c k α ∨ k for α ∨ i ∈ Φ ∨ and c i ∈ Z .By Proposition 3.6 ℓ R ( w ) ≤ k . To show that 2 k is also a lower boundsuppose that w = r r · · · r m with each r i = r α i ,c i ∈ R . Because thereflection r i moves points in the α ∨ i direction and λ has real dimension k (Theorem 3.7), the coroots α ∨ i must span a subspace of dimensionat least k . Next, we may assume that the first k reflections have lin-early independent coroots since it is possible to use Lemma 3.5 to moveany k reflections with linearly independent coroots to the front. Thisalters the reflection factorization of w but leaves the total length un-changed. Let u = r r · · · r k and let v = r k +1 · · · r m so that w = uv , let u = p ( u ) and v = p ( v ) where p is the homomorphism p : W ։ W EFLECTION LENGTH 9 and note that ℓ R ( u ) = k by Proposition 2.5. Since w is a transla-tion, p ( w ) is the identity and v = u − . Finally, by Proposition 2.4we have ℓ R ( v ) ≥ ℓ R ( v ) = ℓ R ( u − ) = ℓ R ( u ) = k , which means thatany reflection factorization of v has length at least k . This implies that m ≥ k and as a consequence every reflection factorization of w haslength at least 2 k . (cid:3) Theorem A (Explicit affine upper bounds) . If W is an affine Coxetergroup that naturally acts faithfully and cocompactly on R n then everyelement of W has reflection length at most n and there exist elementsin W with reflection length equal to n .Proof. Proposition 4.1 shows that 2 n is an upper bound. Proposi-tions 3.4 and 4.3 show that it is optimal. (cid:3) Remark 4.4 (Finite factors) . The cocompactness assumption in The-orem A essentially means that W does not have any finite irreduciblefactors (Remark 2.2). When finite irreducible factors are present, theoptimal upper bound can be lowered accordingly. In particular, if W = W f × W a where W f is the product of the finite irreducible factors, W a is the product of the affine irreducible factors and R n = R n f ⊕ R n a is the orthogonal decomposition preserved by this splitting then byProposition 1.2, Corollary 2.6 and Theorem A the optimal upper boundfor the reflection length function on W is n f + 2 n a = 2 n − n f .We should note that we have not found an elementary way to com-pute ℓ R ( w ) exactly for a generic w in an affine Coxeter group W .Techiques for easily computing ℓ R ( w ) would be useful. For example,they would enable one to investigate whether the polynomial f λ ( x ) = P w ∈ W x ℓ R ( t λ w ) for λ ∈ L has properties similar to f ( x ), the Solomonpolynomial discussed in Proposition 2.8. All we can say at the moment(as a consequence of Proposition 4.1) is that when λ is k -dimensional,the polynomial f λ ( x ) has degree at most k + n and is divisible by t k .5. An unbounded example
When W is a Coxeter group that is neither finite nor affine we con-jecture the following. Conjecture 5.1 (No upper bounds) . The reflection length function ona Coxeter group W has a uniform upper bound if and only if W is ofspherical or affine type. We are not currently able to prove this conjecture but we can showthat it holds in at least one special case. Let W be the free Coxetergroup on three generators, i.e. the group generated by three involutions and with no other relations. Using a criterion of Dyer [3] we can showthat the n -th power of the product of the three standard generators hasreflection length n + 2. In particular, the reflection length function isunbounded on W and Conjecture 5.1 holds in this case. We note that W can be viewed as a reflection group acting on the hyperbolic planegenerated by the reflections in the sides of an ideal triangle. Provingthat all hyperbolic triangle Coxeter groups have unbounded reflectionlength would be first step towards proving Conjecture 5.1. Acknowledgements
We would like to thank Matthew Dyer and JohnStembridge for helpful early conversations about the ideas in this noteand Rob Sulway for conversations about related topics that arise in hisdissertation.
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