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2 Preliminaries
2.1 Ricci curvature of a locally finite graph We recall here the definitions and main facts about the discrete Ricci curvature of a locally finite graph, the main reference for this section is Subsection 1.1 from [21], the notation is slightly different. Let G be a graph, we assume it to be undirected, with no loops and with no multiple edges (i.e. a simple graph); also we ask that it has no isolated vertices. We denote by (G) the set of vertices of G, by (G) its edges and we take δ(x,y) as the function V E δ : (G) (G) N that gives the distance between two vertices. For a fixed V ×V → ∪ {∞} x (G) and i N we define the set: ∈V ∈ B(i, x) := u (G) δ(x, u) = i ; { ∈V | } we denote by d(x) the cardinality of B(1,x) and call it the degree of x.
Definition 1. We say that a given G is locally finite if d(x) < for all x (G). ∞ ∈V
2 From now on we assume G to be locally finite. Given real functions f and g on (G) and x (G) we define the following operators: V ∈V • ∆(f)(x) := (f(v) f(x)); v∈B(1,x) − • Γ(f, g)(x) :=P1 (f(x) f(v))(g(x) g(v)); 2 v∈B(1,x) − − • Γ (f)(x) := 1 (∆(Γ(P f,f))(x)) Γ(f, ∆(f))(x). 2 2 − Instead of Γ(f,f)(x) we write Γ(f)(x). Note that Γ(f)(x) 0 x (G), the equal- ≥ ∀ ∈ V ity holds if and only if f(x) = f(v) for all v in B(1,x). Also note that ∆(f)(x) = f(v) d(x)f(x). v∈B(1,x) − P These definitions allow us, following [21] and [29], to introduce the Ricci curvature of a graph: Definition 2. The discrete Ricci curvature of a graph G, denoted Ric(G), is the maxi- mum value K R such that for any real function f on (G) and any vertex x, ∈ ∪ {−∞} V Γ (f)(x) KΓ(f)(x) . 2 ≥ There is also a local version of this definition: Definition 3. The local Ricci curvature of a graph G at a given point x¯ (G) is the ∈V maximum value K R such that for any real function f on (G) , Γ (f)(¯x) ∈ ∪ {−∞} V 2 ≥ KΓ(f)(¯x) holds. The local curvature so defined is denoted Ric(G)x¯.
We obtain that Ric(G)= infx∈V(G) Ric(G)x. For brevity, in the rest of this work, we often say ”curvature” instead of ”Discrete Ricci curvature”. Remark 4. Note that if c, d R and f : (G) R then ∆(f)=∆(f + c), Γ(f + c, g + ∈ V → d) = Γ(f, g) and Γ2(f + c)=Γ2(f). We may therefore assume that in the expression Γ (f)(x) KΓ(f)(x), f satisfies f(x) = 0. This allow us to use the following formula 2 ≥ for Γ: 1 Γ(f)(x)= f(v)2. 2 v∈XB(1,x) The following result appears in [21, Subsection 1.1]:
Proposition 5. Γ2 can be expressed through the following formula when f(x) = 0:
2 1 2Γ (f)(x)= (f(u) 2f(v))2 + f(v) + 2 2 − u∈XB(2,x) v∈B(1,uX)∩B(1,x) v∈XB(1,x) 1 4 d(x) d(v) + 2(f(v) f(v′))2 + (f(v)2 + f(v′)2) + − − f(v)2, ′ − 2 2 {v,vX}∈E(G) v∈XB(1,x) where the third sum runs over all v, v′ B(1,x) such that v, v′ is an edge in G. ∈ { }
3 The following simple observation does not appear anywhere in the literature so we include its proof:
Lemma 6. We can write Γ (f)(x) Ric(G)= inf 2 (1) x,f Γ(f)(x) where x ranges in (G) and f ranges over the real functions defined on (G) such that V V f(x) = 0 and Γ(f)(x) > 0.
Proof. We know that G has no isolated vertices, so there is a function f¯ on (G) such V that Γ(f¯)(x) > 0. Any K that satisfies
Γ (f)(x) KΓ(f)(x) x, f 2 ≥ ∀ ∀ also satisfies Γ (g)(x) K 2 < x, g s.t.Γ(g)(x) > 0. ≤ Γ(g)(x) ∞ ∀ Let now f˜ : (G) R be such that Γ(f˜)(x) = 0, it comes easily from Proposition 5 V → that in this case Γ (f˜)(x) 0 and so in particular Γ (f˜)(x) KΓ(f˜)(x). The statement 2 ≥ 2 ≥ follows.
Remark 7. From Lemma 6 we deduce the following formula for the local Ricci curvature:
Γ2(f)(x) Ric(G)x := inf , f Γ(f)(x) where f ranges among the functions on (G) such that f(x) = 0 and Γ(f)(x) = 0. V 6 Remark 8. From the formulas for Γ and Γ2 we obtain that Ric(G)x depends only on the subgraph obtained as the union of the paths of length 1 and 2 starting from x. The set of vertices of this graph is x B(1,x) B(2,x) and the edges are the ones connecting { }∪ ∪ vertices in B(1,x) to vertices in x B(1,x) B(2,x). We will refer to such a subgraph { }∪ ∪ as the length-2 path subgraph of x. As a consequence two vertices with isomorphic length- 2 path subgraphs have the same local Ricci curvature.
We conclude with the following remark about triangle-free graphs. These are graphs with no x, u, v (G) such that x, u , x, v , u, v (G): ∈V { } { } { }∈E Remark 9. If G is a triangle-free graph, then: 1 2Γ (f)(x)= (f(u) 2f(v))2+ 2 2 − u∈XB(2,x) v∈B(1,uX)∩B(1,x) 4 d(x) d(v) +( f(v))2 + − − f(v)2. 2 v∈XB(1,x) v∈XB(1,x) We obtain this formula just by erasing the third sum in the equation in Proposition 5.
4 We go on by recalling [21, Theorem 1.2] and a corollary:
Theorem 10. Let G be a locally finite graph, t(v, v′) be the function that counts the num- ′ ′ ber of triangles containing both the vertices v and v and let T = sup{v,v′}⊂V(G)t(v, v ). Then Ric(G) 2+ T . ≤ 2 Corollary 11. Let G be a graph with no triangles, then Ric(G) 2. ≤ Knowing the Ricci curvature of a graph allows one to obtain other information about it. We conclude by presenting some results about the spectral gap and the isoperimetry of a graph that depend on a lower bound on the Ricci curvature, and that are used in this work, our references are [21, Section 3] and [21, Section 4]. To learn more about isoperimetric inequalities for graphs we refer the reader to [18, Section 4]. From now until the end of the section we consider only finite graphs. We start by defining the spectral gap, for a more detailed overview of spectral graph theory the reader can refer to [8].
Definition 12. The adjacency matrix of G is a square matrix of dimension (G) |V |× (G) such that the entry Aij of the matrix is 1 if there is an edge between vertex i and |V | vertex j of G, Aij is 0 otherwise. The Laplacian matrix is defined as:
∆ := D + A − where D is the diagonal matrix whose entries are the degrees of the vertices and A is the adjacency matrix of the graph.
Definition 13. Let G be a graph and ∆ its Laplacian matrix. The spectral gap of G, λG is the least non zero eigenvalue of ∆. − Where no ambiguity occurs we shall just denote the spectral gap as λ. The following results, proved in [21], show how the Ricci curvature of a graph can help bound its spectral gap and how these two values allow an isoperimetric inequality.
Theorem 14. Let G be a graph with Ric(G) > 0, then λ Ric(G). ≥ Theorem 15. Suppose G has Ric(G) K, for some K R 0 . Then, for any subset ≥ ∈ \ { } A (G), ⊂V 1 λ A ∂A min √λ, A 1 | | . | |≥ 2 { 2 K }| | − (G) | | |V | Here, by ∂A, we mean the collection ofp all edges connecting A to its complement. Notice that for Theorem 14 it is crucial to have Ric(G) positive, but positivity is not required for Theorem 15. With a different bound on the spectral gap it is still possible to apply Theorem 15 to families of graphs with a negative lower bound for the Ricci curvature. Finally, we recall the following result that bounds the spectral gap of any Cayley graph:
5 Corollary 16. Let G be a finite group and S G be a set of generators. We denote by ⊂ Cay(S) the Cayley graph of G defined by taking S as set of generators. Then: G λ | | . Cay(S) ≥ d S d | | Where d is the diameter of Cay(S). This previous result is proved in [12].
2.2 Matrices and eigenvalues In Section 3 we need to study the maximum eigenvalue of some matrices. For this purpose the results that are presented in this subsection are of great usefullness. First we recall a result that bounds the eigenvalues of any matrix: n×n n Theorem 17 (Gershgorin). Let M = (mi,j) be a matrix in R , let Ri = mi,j j=1,j6=i | | and P Ki := z C z mi,i Ri . { ∈ || − |≤ } n Then all the eigenvalues of M are contained in Ki. The Ki are called Gershgorin ∪i=1 circles. A proof of Gershgorin’s Theorem can be found, e.g., in [15, Subsection 7.2.1] (see also [14]). We now recall some formulas for the eigenvalues of some particular families of ma- trices. Let us consider the following matrix: α + b c − . a b .. 0 . . . R5 M(α, β, a, b, c) = ...... (α, β, a, b, c) . ∀ ∈ .. 0 . b c a β + b − Its eigenvalues can be computed using the next result first proved in [26, Theorem 4]: Theorem 18. Suppose that ac > 0 and that α = β = √ac = 0, then the eigenvalues 6 λ1, . . . , λn of M(α, β, a, b, c) are given by iπ λ = b + 2 αβ cos( ), i = 1,...,n. i n Finally we consider one last familyp of matrices:
c0 cn−1 ... c2 c1 c c c c 1 0 n−1 2 . .. . n M(c ,...,cn )= . c c . . (c ,...,cn ) R . 0 −1 1 0 0 −1 ∈ .. .. cn−2 . . cn−1 cn cn ... c c −1 −2 1 0 6 These matrices are such that each descending diagonal from left to right is constant (matrices with this property are called Toeplitz matrices, see e.g. [17]). Furthermore the values on the diagonals are repeated as pictured, in this case we call these circulant Toeplitz matrices. The following theorem ([17, Section 3.1]) holds:
Theorem 19. The eigenvalues λ1, . . . , λn of M(c0,...,cn−1) are given by
j 2j (n−1)j λj = c + cn ζ + cn ζ + . . . + c ζ , j = 0, 1,...,n 1. 0 −1 −2 1 − 2iπ Where ζ = e n .
2.3 Coxeter groups In this subsection we recall the definition and main facts about Coxeter groups, the main reference is [4]. Our main goal is to define a family of graphs associated to Coxeter groups, namely the weak order graphs ( denoted V (W )). Coxeter systems are pairs (W, S) where W is a group generated by the elements in S and S = si i I is a finite set with the following relations: { } ∈
mi,j (sisj) = e.
The values mi,j are usually seen as the entries of a symmetric matrix with mi,i = 1 for all i I and mi,j 2 (including mi,j = ) for all i, j I, i = j. W is called a Coxeter ∈ ≥ ∞ ∈ 6 group, S turns out to be a minimal set of generators, its elements are called Coxeter generators. All the information about a Coxeter group can be encoded in a labeled graph called the Coxeter graph. Its set of vertices is S, and there is an edge between two vertices si and sj if mi,j 3, such an edge is labeled with mi,j if mi,j 4. We state ≥ ≥ a very classical result from Coxeter Theory, namely the classification of finite Coxeter groups, for more details the reader can see [4, Appendix A1] and [19, Chapter 2].
Theorem 20. Given a finite Coxeter group W , this can be written in a unique way as direct product of the following irreducible Coxeter groups:
W = W . . . Wk 1 × ×
With Wi of the following kind:
• An n 1; ≥ • Bn n 2; ≥ • Dn n 3; ≥ • I (m) m 2; 2 ≥ • H3, H4;
• E6, E7, E8;
7 • F4.
Remark 21. Groups of type An are symmetric groups, in particular An = Sn+1. The groups of type Bn, called hyperoctahedral groups, are the groups of permutations π of the set 1,..., n such that π(a) = π(a). Finally Dn is the subgroup of Bn of {± ± } − permutations such that an even number of positive elements has negative image. Dn is called even hyperoctahedral group. We define strictly linear Coxeter systems following [27]:
Definition 22. Given a Coxeter system (W, s ,...,sn ), this is called strictly linear { 1 } if:
• mi,j 3 if i j = 1; ≥ | − | • mi,j = 2 if 1 < i j ; | − | Notice that these groups are the ones whose Coxeter graph is a path. We continue with some classical definitions in Coxeter theory. Given an element w in W , this can be written as a product of elements in S
w = s1 ...sk. If k is the minimal length of all the possible expressions for w, we say that k is the length of w and we write ℓ(w)= k. We define in W the set of reflections as the union of all the −1 conjugates of S, T := w W wSw . The definitions of length and reflections allow us ∪ ∈ to define two partial orders on the set W : Definition 23. Given w W and t T , if w′ = tw and ℓ(w′) <ℓ(w) we write w′ w. ∈ ∈ → Given two elements v, w W we say that v w according to the Bruhat order if there ∈ ≥ are w , . . . , wk W such that 0 ∈ v = w w . . . wk wk = w. 0 ← 1 −1 ← Thus we obtain a first partial order on W . The undirected Bruhat graph associated to a Coxeter group, denoted B(W ), is the graph whose set of vertices is W and such that there is an edge between two vertices w, v if and only if w v or v w. → → The second order we define on the elements of a Coxeter group is the weak order on (W, S):
Definition 24. Given u, v W , we say that u R w if w = us ,...sk, for some si S ∈ ≤ 1 ∈ such that ℓ(us ...si)= ℓ(u)+ i for every 1 i k. 1 ≤ ≤ This is the definition of the right weak order on W , analogously one defines the left weak order, multiplying the s1,...sk on the left. These two orders are isomorphic via the map w w−1. → Given a Coxeter group (W, S) we denote by V (W ) the Hasse graph associated to the weak order, we call it simply ’weak order graph’. This has set of vertices W , two elements in the graph are connected by an (undirected) edge if one covers the other according to the weak order.
8 3 Eigenvalues and discrete Ricci curvature
What follows can be seen as a sequel of Subsection 2.1, after introducing Ricci curvature for graphs here we present some new results on its computation. The main result is 27 which describes a method to recover the local Ricci curvature of a graph by computing the minimal eigenvalue of a given matrix. Among the consequences of Theorem 27 we find some inequalities to bound the value of discrete Ricci curvature for graphs which satisfy some conditions. These conditions are, as seen on Theorem 10, about the number of triangles of the graph or, similarly, on the number of quadrilaterals. First we notice that graph automorphisms respect the Ricci curvature:
Lemma 25. If G is any locally finite graph and χ is a graph automorphism then the following holds: Ric(G)x = Ric(G)χ(x). Proof. Trivial from Remark 8.
Now we introduce our main result about the computation of discrete Ricci curvature: We fix x (G), for any v B(1,x) we denote by v := B(2,x) B(1, v) and by ∈ V ∈ U ′ ∩ ′ tv the number of triangles containing vertices v and x, so tv = v B(1,x) v, v |{ ∈ |{ } ∈ (G) . For any u B(2,x) we denote by nu the cardinality of B(1, u) B(1,x), E }| ∈ ∩ this is the number of paths of length 2 joining x and u. We define a function T : B(1,x) B(1,x) 0, 1 as follows: × → { } 1 if v, v′ (G) T (v, v′)= { }∈E . (0 otherwise
Definition 26. Given a locally finite graph G and a vertex x of G. Assume that d(x)= B(1,x) = d, and let us define an order on the elements adjacent to x that makes | | B(1,x)= v , . . . , vd . We define the following matrix associated to x: { 1 }
2(nu−1) 4−d(x)−d(vi ) 3 +1+ + tvi if i = j u∈Uvi nu 2 2 Aij(x)= , 2 T v , v i j ( u∈Uv ∩Uv ( n )+1+2 ( i j) if = P i j − u 6 P for all i, j 1,...,n . ∈ { } An example of the construction of such a matrix can be found in Example 28.
Theorem 27. Given a locally finite graph G and x a vertex of G, then
Ric(G)x = min λ λ is an eigenvalue of A(x) . { | } As a consequence
Ric(G) = inf λ λ is an eigenvalue of A(x),x (G) . { | ∈V }
9 Proof. Let u B(2,x), we notice that ∈ 2 2 2 (f(u) 2f(v)) = nuf(u) + 4f(v) − − v∈B(1,xX)∩B(1,u) v∈XB(1,x) 4f(u) f(v) v∈B(1,xX)∩B(1,u) 2 2 = √nuf(u) f(v) + − √n v∈XB(1,x) 4 (f(v) f(v′))2 n u ′ − {v,v }⊂BX(1,x)∩B(1,u) 4 (f(v) f(v′))2. n ≥ u ′ − {v,v }⊂BX(1,x)∩B(1,u) As f varies among the functions on (G) such that f(x) = 0 and Γ(f)(x) = 0. The V 6 minimal value of (f(u) 2f(v))2 is achieved for those f satisfying f(u)= v∈B(1,x)∩B(1,u) − 2 f(v). nu v P Therefore P p(x) + ( f(v))2 + (4−d(x)−d(v)) f(v)2 Ric G v∈B(1,x) v∈B(1,x) 2 x( ) = inf 2 f P v∈B(1,xP) f(v)
P ′ 2 1 2 ′ 2 ′ ′ [2(f(v) f(v )) + (f(v) + f(v ) )] v,v ∈B(1,x),{v,v }∈E(G) − 2 . + 2 P ( v∈B(1,x)f(v) f(v)) Where P 2 p(v)= (f(v) f(v′))2. n ′ u − u∈XB(2,x) v6=v ∈B(1X,u)∩B(1,x) We denote the d elements of B(1,x) as v1, . . . , vd. We notice that only the d values f(v), v B(1,x) appear in this last formula. We ∈ see them as d variables x ,...,xd R where xi = f(vi) and study the minimum of this 1 ∈ rational function. Both numerator and denominator are homogeneous polynomials of degree two. There- n 2 fore we can assume i=1 xi = 1. The numerator can be seen as the polynomial associ- ated to a quadratic form, the matrix associated to such a quadratic form is the matrix P A(x). This can be easily proved computing the coefficient of xixj in the formula for Ric(G)x and see that it coincides with A(x)ij. We obtain t Ric(G)x = inf y A(x)y. y∈Sd−1 From the definition of A(x) we notice that it is symmetric therefore all its eigen- values are real, in particular we can talk about the minimum eigenvalue of A and call
10 it λmin(x). We deduce that Ric(G)x = λmin(x) and conclude knowing that Ric(G) = infx∈V(G) Ric(G)x.
Example 28. Let G be the Bruhat graph of S3 = A2 s s s 12 •23 12
s23s12 • •s12s23
s12 • •s23
• e
We fix x = e, in this case we have that:
B(1,x)= s ,s ,s s s ; B(2,x)= s s ,s s d(x) = 3. { 12 23 12 23 12} { 23 12 23 12} Notice that all the vertices x in G have isomorphic length-2 path subgraphs, therefore A(x)= A(x′) for all x,x′ (B(W )) = W . We take x = e, we have ∈V
nu = 2 for any u B(2,x) ∈ and v = B(2,x) for any v B(1,x). U ∈ The graph is triangle free so tv and Tv,v′ are zero. The resulting matrix is
8 1 1 3 3 3 1 −8 − 1 A(e)= 3 3 3 − 1 1 −8 − 3 − 3 3 10 whose eigenvalues are 2 and 3 (the latter with multiplicity two). We obtain that Ric(G) = 2.
Theorem 27 is a helpful tool to compute the Ricci curvature of a finite graph. We now show how combining this result with some theorems on the eigenvalues of a matrix we can obtain inequalities for the curvature of infinite families of graphs.
Theorem 29. Let G be a triangle free graph then
3d(x)+ d(y) Ric(G) 4 max ≥ − x,y∈E(G) 2
11 Proof. Let x (G) and d = d(x), we write B(1,x)= y ,...,y . Let yi = y be the ∈V { 1 d(x)} element in B(1,x) with maximum degree. We want to compute the Gershgorin circle (see Theorem 17) associated to the i-th column of A(x). Its center is
2(nu 1) 4 d(x) d(y) Ai,i(x)= − +1+ − − , nu 2 u∈B(2,xX)∩B(1,y) and its radius is:
d(x) d(x) 2 Ai,j(x) = + 1 . | | |− nu | j ,j i j ,j i =1X6= =1X6= u∈B(1,y)∩BX(1,yj )∩B(2,x) The minimal real value contained in such a circle is
d(x) d(x) 2 A , Ai, A , ( +1) = 1 1 − | 1|≥ 1 1 − |− n | i i u X=2 X=2 u∈B(1,y)∩BX(1,yi )∩B(2,x) 2(nu 1) 4 d(x) d(y) = − +1+ − − nu 2 − u∈B(2,xX)∩B(1,y) 2(nu 1) − d(x)+1= − nu − u∈B(2,xX)∩B(1,y) 4 d(x) d(y) + 2 2d(x) + 2 = − − − 2 3d(x) d(y) =4 − . − 2 We conclude that the minimal eigenvalue of A(x) is greater than 4 3d(x)+d(y) . From − 2 Theorem 27 we know that 3d(x)+ d(y) Ric(G) = min λ λ is eigenvalue of some A(x) min 4 . { | }≥ x∈V(G),y∈B(1,x){ − 2 } The claim follows.
We go on by considering a corollary of Theorem 29 that holds for graphs with no triangles and no quadrilaterals. These are graphs such that given a path of length 4 v v v v v the five vertices are distinct. With this condition any length-2 path 1 − 2 − 3 − 4 − 5 subgraph has no cycles. Corollary 30. Let G be a graph with no triangles nor quadrilaterals. The following inequality then holds for Ric(G): 3d(x)+ d(v) 2+ d(x′) d(v′) 4 max Ric(G) min(2, − ) − {x,v}∈E(G) 2 ≤ ≤ 2 where x′, v′ are adjacent vertices that minimizes d(x′) d(v′). −
12 Proof. We fix a vertex x and u B(2,x), we notice that nu = 1. For the same reason ′ ∈ v v′ = for every (v, v ) pair of distinct elements in B(1,x). Suppose that B(1,x) = U ∩U ∅ | | n, we assume that these n elements are ordered B(1,x) = vi i ,...,n. Therefore the { } =1 matrix A(x) described in Theorem 27 is:
d1 + 1 1 . . . 1 . . . 1 .. .. . A x , ( )= . . . . .. .. 1 1 . . . 1 dn + 1 4−d(x)−d(vi ) where di = 2 . We know that Ric(G)x is the minimum eigenvalue of A(x). We apply Gershgorin’s Theorem to this matrix to bound this value. First we consider the Gershgorin circles, these are the sets
Ki = z C : z di 1 d 1 . { ∈ | − − |≤ − } 4−d(x)−d(v) 8−3d(x)−d(v) We deduce that Ric(G)x 2 +1 d+1= 2 where v is the element in ≥ − 4−d(x)−d(v) B(1,x) with maximum degree. Also, we obtain that Ric(G)x 2 + d(x) 1= 2+d(x)−d(v) ≤ − 2 where v is the element in B(1,x) with minimum degree. Notice that because G is triangle free we know from 9 that Ric(G) 2. We obtain the following bounding ≤ for Ric(G): 3d(x) d(v) 2+ d(x′) d(v′) 4+ − − Ric(G) min(2, − ) 2 ≤ ≤ 2 where (x, v) is a pair of adjacent vertices that maximizes 3d(x) d(v) and x′, v′ are − adjacent vertices that minimizes d(x′) d(v′). − This result can be used to bound the Ricci curvature of any tree . A new corollary follows, this one gives the exact Ricci curvature of a subfamily of graphs with no triangles nor quadrilaterals. Corollary 31. Let G be a graph with no triangles and quadrilaterals and with the prop- erty that all the length-2 path subgraphs are isomorphic. Then
Ric(G) = 2 d − where d is the degree of any vertex in G. Proof. We fix any vertex x in G, from the hypothesis on the length-2 path subgraphs we can say that Ric(G)x = Ric(G). The matrix A(x) associated to this vertex is: 3 d 1 . . . 1 − . . . 1 .. .. . A x , ( )= . . . . .. .. 1 1 . . . 1 3 d − 13 We conclude that A(x) is a circulant matrix and Ric(G) is its smallest eigenvalue. By Theorem 19 the eigenvalues of A(x) are:
d−1 k λk = 3 d + ζ i = 2 d ζ any primitive d-th root of unity for k = 1,...,d 1 − − − i X=1 λ = 3 d + d 1 = 2. 0 − − By Theorem 19 we conclude that Ric(G) = 2 d. − This Corollary implies the following:
Corollary 32. Any integer smaller than 2 is the discrete Ricci curvature of a graph.
In particular given z Z, z < 2 it is the curvature of the Cayley graph of the free ∈ group generated by 2 z elements. − 4 Ricci curvature of Weak orders
In this section we compute the Ricci curvature of the Hasse graphs of the weak Bruhat order associated to Coxeter groups, according to our notation we write V (W ) referring to the weak order graph of W . The study of the curvature of these weak order graphs, together with Theorem 15 allows us to obtain an isoperimetric inequality. Furthermore the weak order graph associated to affine Weyl groups is locally finite as well, this leads to two theorems: one for finite Coxeter groups (33) and one for affine Weyl groups (37). The main idea behind the proof of the result in this Section is similar to the one used to prove Theorem 27: we find the minimum of a rational function studying the minimal eigenvalue of a matrix. We are ready for the first theorem of this section, that only deals with finite irre- ducible Coxeter groups. A way to study the Ricci curvature of V (W ) if W is a finite non-irreducible Coxeter group is presented in Theorem 34.
Theorem 33. Given (W, S) a finite irreducible Coxeter system and V (W ) the weak order graph associated to it, the following holds:
• Ric(V (W )) = 2 cos( π ) if W is a strictly linear Coxeter group; − |S|
• 4 Ric(V (W )) 2 if W = Dn with n 4 and Ric(V (D )) 1.7; − ≤ ≤− ≥ 3 ≃− • Ric(V (E )) 2.30, Ric(V (E )) 2.33 and Ric(V (E )) 2.34. 6 ≃− 7 ≃− 8 ≃− Proof. Given an element g W we can consider the automorphism induced on V (W ) ∈ by the multiplication by g. By Lemma 25 the local Ricci curvature at a given vertex (say e) coincides with the global Ricci curvature of V (W ). We compute the former, therefore we consider the length-2 path subgraph with center e. First we want to study how many paths connect a given element in u B(2, e) to e. Because every edge in ∈
14 V (W ) is labeled with a simple rlefection, the study of the paths correspond to the study of the equalities ss′ = zz′ with s,s′,z,z′ S. Note that if s s = s s = e in W with ∈ 1 2 3 4 6 s ,s ,s ,s S and (s ,s ) = (s ,s ) then s = s , s = s and m , = 2. Let WI 1 2 3 4 ∈ 1 2 6 3 4 1 4 2 3 1 2 and WJ the parabolic subgroups of W generated by I = s ,s and J = s ,s . We { 1 2} { 3 4} notice that WI WJ = 0 , from WI WJ = WI J it follows from [4, Proposition 2.4.1] ∩ 6 { } ∩ ∩ that I J = , this implies that either s = s , either s = s . From this it follows that ∩ 6 ∅ 1 4 2 3 s1 = s4 and s3 = s2. A vertex in B(2, e) is therefore connected to e through one or two paths. Here follows the picture of the length-2 path subgraphs of e in the weak order associated to S4. s s 12• 23
•s12 •s23s12
s12s34 • e• •s23
•s34 •s23s34
• s34s23
′ If there is one path from u to e, say u v e, then u = vv and mv,v′ 3; if there ′ − − ′ ≥ are two, say u v e and u v e, then u = vv and mv,v′ = 2. − − − − Let u B(2, e), consider a pair of reflections in S, say s and s′, we then have the ∈ following inequalities:
(f(u) 2f(s))2 + (f(u) 2f(s′))2 =2f(u)2 + 4f(s)2 + 4f(s′)2 4f(u)f(s) − − − − 4f(u)f(s′) + 4f(s)f(s′) 4f(s)f(s′) − =2(f(u) f(s) f(s′))2 + 2(f(s) f(s′)) − − − 2(f(s) f(s′))2 ≥ − If u = ss′ = s′s and
(f(u) 2f(s))2 0 if u = ss′, with ss′ = s′s. − ≥ 6 ′ Therefore the minimal values for Γ2(f)(e) are reached for f(u) = f(s)+ f(s ) if u = ss′ = s′s and u = 2f(s) otherwise. We substitute these minimal values in the formula
15 presented in Proposition 5 for the discrete curvature obtaining: 1 (f(u) 2f(s))2 + ( f(s))2 + (2 d(e)) f(s)2 V W 2 s∈S,u∈B(1,s)∩B(2,e) − s∈S − s∈S Ric( ( )) := inf 2 f P s f(sP) P ′ 2 2 2 {s,s′}∈S(f(s) f(s )) + ( s∈S f(s)) + (2 d(e)) s∈S f(s) =inf − P − f f s 2 P Ps∈S ( ) P ′ 2 ′ (f(s) f(s )) {s,s }⊂S,ms,s′ ≥3 − P 2 − P s∈S f(s) ′ 2 ′ (f(s) f(s )) {s,sP}⊂S,ms,s′ ≥3 − . =2 sup 2 (2) − f P s∈S f(s) We focus now on the rationalP function that appears in equation (2). Due to the condition f(s)2 = 0 we can simply study s∈S 6 2 2 P PW ( xs s S) := (xs xs′ ) with x = 1. { } ∈ − s ′ s∈S {s,s }⊂XS,ms,s′ >2 X
The polynomial PW (x) can be seen as the coordinate representation of the bilinear form induced by the matrix MW with
1 if sisj = sjsi − 6 [MW ]i,j = 0 if sisj = sjsi and i = j (3) 6 # k sisk = sksi if i = j. { | 6 } The matrix MW is symmetric, therefore diagonalizable and so has a maximum eigenvalue λW R. We notice that the structure of MW does not depend on the order mi,j of sisj ∈ but rather on whether mi,j is different or not from 2. This information is encoded by the Coxeter graph of W , ignoring the labels of the edges. In particular we notice that the Matrix MW has the same zero entries of the Cartan matrix associated to W . We split the proof in three cases as follows: case 1 W is a strictly linear finite Coxeter group; case 2 W is of type Dn, n 3; ≥ case 3 W is of type E6, E7 or E8. Case 1. We can see P as a coordinate representation of a quadratic form in Rn induced by the following matrix: 1 1 0 . . . 0 − . . . 1 2 .. .. . − . . . MV (W ) = 0 ...... 0 . .. .. . . . 2 1 − 0 . . . 0 1 1 − 16 By Theorem 18 applied with
α = β = 1, b = 2, a = c = 1 − we conclude that the eigenvalues of MV (W ) are iπ λ = 2(1 + cos( )) for i = 1he...,n. i n π The maximum eigenvalue is therefore 2(1 + cos( n )). So π π Ric(V (W )) = 2 2(1 + cos( )) = 2 cos( ). − n − n where n = S . | | Case 2. With the same procedure we find the matrix associated to the groups of type Dn, this is: 1 0 1 − 0 1 1 − 1 1 3 1 − − − M = 1 2 1 Dn − − . . . ...... 1 2 1 − − 1 1 No formulas seem to be known for the eigenvalues of these− matrices, but we can bound the greatest one with the help of Gershgorin’s Theorem ( Theorem 17). The Gershgorin circles are: z C : z 1 1 for i = 1, 2,n; { ∈ | − |≤ } Ki = z C : z 3 3 for i = 3; { ∈ | − |≤ } z C : z 2 2 otherwise. { ∈ | − |≤ } n We notice that Ki =K , so λ 6. For n = 3 and n = 4 it is easy to compute ∪i=1 4 max ≤ the minimum eigenvalue of MDn , it is 3, 7.... and 4 respectively. For n 5 we can take T T ≥ e + en = [0, 0, 1, 0,..., 1] and see that (e + en) MD (e + en)=4, so 4 λ 6. 3 3 n 3 ≤ max ≤ Therefore 4 Ric(WD ) 2 for n 4 and Ric(WD )= 1.7.... for n = 3 . − ≤ n ≤− ≥ 3 − Case 3. For these exceptional groups it is sufficient to study the maximum eigenvalue of the following matrices:
1 10 0 0 0 − 1 2 0 1 0 0 − − 0 0 1 1 0 0 ME6 = − 0 1 1 3 1 0 − − − 0 0 0 1 2 1 − − 0 0 0 0 1 1 −
17 1 10 0 0 0 0 − 1 2 0 10 0 0 − − 0 0 1 10 0 0 − ME = 0 1 1 3 1 0 0 7 − − − 0 0 0 1 2 1 0 − − 0 0 0 0 1 2 1 − − 0 0 0 0 0 1 1 − 1 1000000 − 1 2 0 10 0 0 0 − − 0 0 1 10 0 0 0 − 0 1 1 3 10 0 0 ME8 = − − − 0 0 0 1 2 1 0 0 − − 0 0 0 0 1 2 1 0 − − 0 0 0 0 0 1 2 1 − − 000000 1 1 − According to Sage Math (see [28]) the maximum eigenvalues are of these matrices are respectively λE6 = 4, 3082775..., λE7 = 4, 33420053... and λE8 = 4, 34292308... and this proves the claim.
From the Theorem of classification of finite Coxeter groups we know that every finite Coxeter group is a direct product of finite irreducible Coxeter groups. To study the discrete curvature of any graph V (W ), W finite Coxeter group, is therefore sufficient to show how the curvature of these graphs interacts with the direct product. This is described in the following theorem.
Theorem 34. Let (W, S) be any finite Coxeter system with W = W . . . Wk and 1 × × S = S . . . Sk where (Wi,Si) is an irreducible Coxeter group for all i = 1,...,k. Let 1 ∪ ∪ V (W ) be the weak order graph associated to (W, S), then:
Ric(V (W )) = min1≤i≤k Ric(V (Wi)).
Proof. We know that two elements s,s′ S may not commute only if they are contained ∈ in the same Si. Therefore
′ 2 ′ ′ (f(v) f(v )) Ric(G ) = 2 sup vv 6=v v − W f v 2 − f P v∈B(1,e) ( ) k ′ 2 i P vv′ v′v,vv′ S (f(v) f(v )) = 2 sup =1 6= ∈ i − , f v 2 − f P P v∈B(1,e) ( ) where f the sup is over real function on W suchP that f(e)=0 and Γ(f)(x) = 0. We can 6 again reduce to the study of the eigenvalues of a matrix MW , this is a block matrix of the following kind:
18 MW1 0 .. . MW = . .. 0 MWk So we have that the maximum eigenvalue of MW is λW := max1≤i≤kλWi . We have therefore information on the Ricci curvature of the weak order graph of any finite Coxeter group. We can use this to obtain an isoperimetric inequality for these graphs:
Corollary 35. Let (W, S) be a finite Coxeter system with A W a subset of this. Then ⊂ the following isoperimetric inequalities hold for V (W ):
• If no dihedral groups appear in the irreducible decomposition of W :
1 W A ∂A | | A 1 | | ; | |≥ 2 S |T | T 2 Ric(V (W )) | | − W | | | | | | | | p • If at least one dihedral group appears in the irreducible decomposition of W :
1 W A ∂A | | A (1 | | ). | |≥ 2s S |T | T | | − W | | | | | | where T denotes the set of reflections.
Proof. First we notice that a weak order graph is a Cayley graph of a Coxeter group taking all the simple reflections as set of generators. From Corollary 16 we have that λ |W | . We want to use this result to apply Theorem 15. We therefore have V (W ) ≥ |S||T ||T | to study λ min √λ, . { √2 Ric(V (W )) } | | From Theorem 33 and Theorem 34 we know that Ric(V (W )) = 0 if a dihedral group appears in the irreducible decomposition of W , from this the second part of the statement follows. Assume now that the irreducible decomposition of W is dihedral-free. Given an element w W , we know that it can be expressed through one of its minimal expressions ∈ as product of simple reflections. Such a minimal expression has length at most T . It | | follows that W S + S 2 + . . . + S |T | S |T | T . | | ≤ | | | | | | ≤ | | | |
19 It is easy to see that |W | < 1. From Theorem 33 we notice that √2 Ric(V (W )) > 1, |S||T |+1 | | it follows that W W W | | > | | > | | . s S |T |+1 S |T |+1 S |T |+1√2 Ric(V (W )) | | | | | | | | The first part of the statement follows.
In the following corollary we present the same result obtained in Corollary 35 in detail for finite irreducible Coxeter groups.
Corollary 36. The following isoperimetric inequalities hold for the weak order graph of finite irreducible Coxeter systems:
An 1 (n + 1)! A ∂A n+1 A 1 | | ; | |≥ 2 ( 2 ) n+1 π | | − (n + 1)! (n) 2 2 cos( n )
Bn p 1 2nn! A ∂A A 1 | | n 2; | |≥ 2 nn2 n22 cos( π )| | − 2nn! ≥ n
Dn p 1 2n−1n! A ∂A A 1 | | n 3; | |≥ 2 nn(n−1)n(n 1)2√2| | − 2n−1n! ≥ − I2(m) 1 1 A ∂A A 1 | | ; | |≥ 2 2m−1 | | − 2m r
E6 1 518640 A ∂A A 1 | | ; | |≥ 2 36 √ . | | − 518640 12 36 4 8
E7 1 2903040 A ∂A A 1 | | ; | |≥ 2 123636√4.8| | − 2903040 E8 1 696729600 A ∂A 1 | | ; | |≥ 2 3012020√4.8 − 696729600 F4 1 1152 A ∂A A 1 | | ; | |≥ 2 24 | | − 1152 12 24 2√2
H3 p 4 A ∂A A 1 | | ; | |≥ 15 | | − 120 10 2√3 p 20 H4 1 14400 A ∂A A 1 | | . | |≥ 2 60 | | − 14400 30 60 2√2 For the last result of this Section we applyp the same methods used to prove Theorems 33 and 34 to compute the Ricci curvature of weak orders of affine Weyl groups. As already noticed the generators of an affine Weyl group has a finite number of generators, therefore the Hasse diagram associated to the weak order is a locally finite graph.
Theorem 37. Let (W, S) be an affine Weyl group, V (W ) the Hasse graph of the weak order associated to it. Then we have
2π n • Ric(V (A˜n)) = 2 cos( ); n ⌈ 2 ⌉ π • Ric(V (W )) = 2 cos( ), if W = C˜n, F˜ , G˜ , A˜ ; − |S| 4 2 1
• 4 Ric(V (W )) 2 if W = B˜n; − ≤ ≤− • 4 Ric(V (W )) 2 if W = D˜n if n 5 and Ric(V (D˜ )) = 3; − ≤ ≤− ≥ 4 − • Ric(V (E˜ )) 2.414 , Ric(V (E˜ )) 2.36, Ric(V (E˜ )) 2.34. 6 ∼− 7 ∼− 8 ∼− Proof. The first part of the proof of Theorem 33 holds for affine Weyl groups as well. We study therefore the eigenvalues of the matrix MW described in 3 depending on the group W . We divide the rest of the proof in five cases:
1) W = A˜n;
2) W strictly linear affine Weyl group;
3) W = B˜n;
4) W = D˜ n;
5) W = E˜6, E˜7, E˜8.
Case 1 We start with the groups of type A˜n, to study the Ricci curvature of the weak order we consider its Coxeter graph which is s ˜•n
••... •• s˜1 s˜2 s˜n−2 s˜n−1
21 and deduce that Ric(G ˜ ) = 2 λ ˜ where λ ˜ is the greatest eigenvalue of the matrix: An − An An 2 1 1 −. . − 1 .. .. 0 − M = ...... A˜n . . . .. .. 0 . . 1 − 1 1 2 − − This is a circulant Toeplitz matrix with c = 2, c = 1, cn = 1 and c = . . . = 0 1 − −1 − 2 cn−2 = 0. From Theorem 19 we know that its eigenvalues are 2j λj = 2 2 cos 0 j n 1. − n ≤ ≤ − The highest value for λ is reached for j = n and j = n . We conclude that λ = j 2 2 A˜n 2π n 2π n ⌊ ⌋ ⌈ ⌉ 2π n 2 2 cos( ) = 2 2 cos( ) and therefore Ric(G ˜ ) = 2 cos( ), in particular, − n ⌈ 2 ⌉ − n ⌊ 2 ⌋ An n ⌊ 2 ⌋ it is 2 if n is even. − Case 2. C˜n, F˜4, G˜2 and A˜1 are strictly linear Coxeter groups, the curvature for π their weak order graph is computed as for An in Theorem 33 and is 2 cos( ). − |S| Case 3. The case of B˜n leads to the same unlabeled graph as the one associated to Dn, with n 4. We can therefore conclude that 4 Ric(B˜n) 2. ≥ − ≤ ≤− Case 4. About the case D˜ n we can see that for n 5 the matrix associated to its ≥ quadratic form is: 1 0 1 − 0 1 1 − . 1 1 3 .. − − 0 . . .. 2 .. M = ...... D˜ n . . . .. .. . 2 . . .. 3 1 1 0 − − 1 1 0 − 1 0 1 − Its Gershgorin’s circles are : z C z 1 1 for i = 1, 2,n 1,n { ∈ || − |≤ } − Ki = z C z 3 3 for i = 3,n 2 { ∈ || − |≤ } − z C z 2 2 otherwise. { ∈ || − |≤ } By Gershgorin’s Theorem we can give an upper bound to λ , namely that it is D˜ n smaller than 6. We take now v = [0, 0, 1, 0,..., 0, 1] Rn, then we have that ∈ vT M v = 4. D˜ n
22 So, as in the case of B˜n we conclude that 4 Ric(V (D˜ n)) 2 if n 5 − ≤ ≤− ≥ We study now the case of Ric(W (D˜4)) the associated matrix is the following:
1 0 1 0 0 − 0 1 1 0 0 − M = 1 1 4 1 1 D˜4 − − − − 0 0 1 1 0 − 0 0 1 0 1 − whose eigenvalues are [5, 1, 1, 1, 0]. We conclude that Ric(V (D˜ )) = 3. 4 − Case 5. To study E˜6, E˜7 and E˜8 it is sufficient to study the eigenvalues of the following matrices:
1 10 0 0 0 0 − 1 2 10 0 0 0 − − 0 1 3 1 0 1 0 − − − M = 0 0 1 2 1 0 0 E˜6 − − 0 0 0 11 0 0 − 0 0 10 0 2 1 − − 0 0 0 0 0 1 1 − 1 1000000 − 0 1 2 10 0 0 0 − − 0 0 1 2 10 0 0 − − 0 0 1 3 1 0 0 1 M ˜ = − − − − E7 0 0 0 1 2 1 0 0 − − 00 0 0 1 2 1 0 − − 00 0 0 0 1 1 0 − 0 0 0 10 0 0 1 − 1 10000000 − 1 2 1000000 − − 0 1 3 10 0 0 0 1 − − − 0 0 1 2 10 0 0 0 − − M = 0 0 0 1 2 10 0 0 E˜8 − − 0 0 0 0 1 2 1 0 0 − − 0 0 0 0 0 1 2 1 0 − − 000000 1 1 0 − 0 0 1000010 − The result follows.
23 Acknowledgement
The author would like to thank Francesco Brenti for suggesting her this problem and for many useful discussions. Furthermore she acknowledges the MIUR Excellence De- partment Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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