A Generalization of Rota's NBC Theorem

We generalize Rota's theorem characterizing the Mobius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak Bruhat order of a nite Coxeter group and the Tamari lattices. We also give a direct proof of the fact that in the geometric case any total order of the atoms can be used. Simple involutions are used in both proofs. Finally we show how involutions can be used in similar situations, speci cally in a special case of Rota's Crosscut Theorem as well as in related proofs of Walker on Hall's Theorem and Reiner on characteristic and Poincar e polynomials. 1 Rota's theorem and its generalization One of the most beautiful and useful theorems in algebraic combinatorics is Rota's theorem [14] characterizing the Mobius function of a geometric lattice in terms of subsets of atoms which are NBC, i.e., contain no broken circuit. In this note we will generalize Rota's theorem to any lattice satisfying a simple condition and give applications to the weak Bruhat order of a Coxeter group and the Tamari lattices. The proof of Rota's theorem is an easy application of the simplest version of the Involution Principle of Garsia and Milne [6]. We also use an involution to show directly that in the geometric case the number of NBC sets is the same for any total ordering of the atoms. Finally we discuss a related proof for a special case of Rota's Crosscut Theorem as well as proofs of Walker concerning the Mobius function as a reduced Euler characteristic and of Reiner connecting characteristic and Poincar e polynomials. We rst review Rota's original theorem. Let L be a nite poset with minimal element 0̂. The M obius function of L is the function : L ! Z (Z being the integers) which is uniquely de ned by X y x (y) = 0̂x (1) where the right side is the Kronecker delta. In particular, if L is the lattice of divisors of an integer then is the number-theoretic Mobius function. Suppose that L is a lattice and let ^ and _ denote the meet (greatest lower bound) and join (least upper bound) operations, respectively. Let A(L) be the set of atoms of L, i.e, all a 6= 0̂ such that there is no x 2 L with 0̂ < x < a. We say that L is atomic if every x 2 L is a join of atoms. Assume further that L is ranked with rank function , which means that for all x 2 L the quantity (x) = length of a maximal 0̂ to x chain is well de ned (independent of the chain). Such a lattice is semimodular if (x ^ y) + (x _ y) (x) + (y) for all x; y 2 L. It is easy to prove, using this inequality and induction, that if B A(L) then (_B) jBj where the vertical bars denote cardinality. So de ne B to be independent if (_B) = jBj and dependent otherwise. If B is independent then we say it is a base for x = _B. If C is a minimal (with respect to inclusion) dependent set then we say that C is a circuit. Now put a total order on A(L) which we will denote to distinguish it from the partial order in L. A circuit 1 ŝ 1 ss a ŝ 0 s c s HHHH HHHH s b s s d @@@@@@@@ P P P P P P P P P P P P Figure 1: An example lattice L C has corresponding broken circuit C = C n c where c is the smallest atom in C. Finally, B A(L) is NBC if it contains no broken circuit. Note that such a set must be independent. Rota's theorem can now be stated. Theorem 1.1 (Rota) Let L be a geometric (i.e., atomic and semimodular) lattice. Then for any total ordering of A(L) we have (x) = ( 1) (x)(number of NBC bases of x): (2) To generalize this result to lattices, we rst need to rede ne some terms since L may no longer be ranked. Call B A(L) independent if _ B < _B for any proper subset B of B. Thus if C is dependent then _ C = _C for some C C. Note that it follows directly from the de nitions that a superset of a dependent set is dependent, or equivalently that a subset of an independent set is independent. The de nitions of base, circuit and broken circuit can now be kept as before. If C is a circuit, it will be convenient to adopt the notation C = C n c for the corresponding broken circuit. This done, our generalization is as follows. Theorem 1.2 Let L be a nite lattice. Let be any total ordering of A(L) such that for all broken circuits C = C n c we have _ C = _C: Then for all x 2 L we have (x) =XB ( 1)jBj (3) where the sum is over all NBC bases B of x. Before presenting the proof, let us do an example. Consider the lattice L in Figure 1 with the atoms ordered a b c d. The circuits of L are fa; b; dg, 2 fa; c; dg and fb; c; dg with corresponding broken circuits fb; dg and fc; dg. It is easy to verify that these circuits satisfy the hypothesis of Theorem 1.2. Also, the element x = 1̂ has two NBC bases, namely fa; dg and fa; b; cg. It follows that (1̂) = ( 1)2 + ( 1)3 = 0 which is readily checked from the de nition of the Mobius function. Proof (of Theorem 1.2). Let ~ (x) =XB ( 1)jBj: Then since (1) uniquely de nes , it su ces to show thatPy x ~ (y) = 0̂x: If x = 0̂ then both sides of this equation are clearly equal to 1. So we assume that x > 0̂ and show that X y x ~ (y) = 0: (4) Consider the set S = fB : B is a base for some y xg with sign function (B) = ( 1)jBj: Clearly PB2S (B) is the left side of (4), so to prove this identity it su ce to nd a sign-reversing involution on S. Let a0 be the smallest atom under x. De ne a map : S ! S by (B) = B 4 a0 where 4 is the symmetric di erence operator. This is clearly a sign-reversing involution as long as it is well-de ned, i.e., as long as B NBC implies (B) NBC. There are now two cases. If (B) = B n a0 then (B) is still NBC because it is a subset of B. Otherwise let B := (B) = B [ a0 and suppose B contains broken circuit C = C n c. If a0 = 2 C then C B contradicting B being NBC. If a0 2 C then we must have c a0 (5) because of the way circuits are broken. But now, using the theorem's hypothesis, c _C = _ C _ B x: Thus c a0 since a0 is the least atom under x, contradicting (5). Note that when L is geometric, then all NBC bases of a given x 2 L have the same number of elements, namely (x). Thus the right sides of (2) and (3) do 3 really coincide in this case. Furthermore, the hypothesis of Theorem 1.2 explains why any ordering of A(L) works in Rota's Theorem: _ C = _C for any C obtained by removing a single atom from the circuit C. On the other hand, it would be nice to have a direct proof of this fact using involutions, which we present next. Proposition 1.3 Let L be a geometric lattice and let O1 and O2 be two total orderings of A(L). Then for all x 2 L we have number of NBC bases of x in O1 = number of NBC bases of x in O2: (6) Proof. It is enough to prove the proposition in the case where O2 is obtained from O1 by transposing the order of two atoms c 1 d adjacent in O1. For i = 1; 2 let BCi (respectively, NBCi) be the set of broken circuits (respectively, NBC bases of x) in the two orders. If C is any circuit containing both c; d and no a smaller than both then C1 = C n c 2 BC1 and C2 = C n d 2 BC2: (7) Any other circuit gives rise to C = C n a 2 BC1 \ BC2: (8) Now de ne a map f : NBC1 n NBC2 ! NBC2 n NBC1 by f(B) = B 4 fc; dg: We must show that f is well-de ned. First note that B cannot contain any broken circuit of the form (8) since B 2 NBC1, but it must contain one of the form (7) since B 62 NBC2. So for some circuit C we have B C2 but B 6 C1, since B 2 NBC1. Thus we must have c 2 B and d 62 B and so f(B) = (B n c) [ d = B [ d (9) where B = B n c. We claim that f(B) 62 NBC1. Indeed C1 = ( C2 n c) [ d and C2 B. Thus equation (9) yields C1 f(B). Now we claim that f(B) 2 NBC2. Suppose not. But f(B) can't contain a broken circuit of the form (7) since c 62 f(B) and c 2 C2 for all C of this type. Thus f(B) must contain a broken circuit of type (8), say f(B) D. However, B 6 D and so by (9) we can write D = D0 [ d 4 where D0 B. Also D = D na where a d (in both orders) since d 2 D. In fact a is smaller than every atom in C since no element of C is smaller than c; d. Write D = D0 [ a [ d and C = C2 [ d where the unions are disjoint. The fact that L is geometric implies that there exists a circuit E C [D which does not contain d 2 C \D. Now if a 62 E then E D0 [ C2 B which contradicts B 2 NBC1. On the other hand, if a 2 E then E = E n a since a is minimal in C [D. But then E B contradicting B 2 NBC1 again. This nal contradiction nishes the proof that f(B) 2 NBC2 and that f is well-de ned. Using precisely the same reasoning, one shows that f has a well-de ned inverse f 1 : NBC2 n NBC1 ! NBC1 n NBC2 given by f 1(B0) = B04 fc; dg = (B0 n d) [ c: Thus f is a bijection and this proves the theorem. Of course, this could also be expressed in terms of an involution : NBC1 4NBC2 ! NBC1 4 NBC2 where (B) = B 4 fc; dg. A related result is the fact that the Tutte polynomial [15] as de ned by external and internal activities is order-independent. Tutte's original proof of this fact in the paper just cited is quite involved. It would be interesting to nd an easier proof using involutions. 2 Applications We now give two examples of lattices which are not geometric, but whose Mobius functions can be computed using Theorem 1.2. We rst note a general result that follows from our main theorem. Corollary 2.1 Let L be a nite lattice such that A(L) is independent. Then the Mobius values of L are all 0 or 1. Speci cally, if x 2 L then (x) = ( ( 1)jBj if x = _B for some B A(L), 0 else. Proof. If A(L) is independent then so is any B A(L). Furthermore, there are no circuits so any such B is NBC. Finally, independence of A(L) implies that _B 6= _B0 for any B 6= B0. The corollary now follows from Theorem 1.2. 5 We note that Corollary 2.1 also follows easily from a special case of Rota's Crosscut Theorem [14], proved by involutions in Section 3. We now derive the Mobius function of the weak Bruhat order of a Coxeter group which is a result of Bjorner [2]. (We do not consider the strong ordering because it is not a lattice in general.) Any terminology from the theory of Coxeter groups not de ned here can be found in Humphreys' book [10]. Let (W;S) be a nite Coxeter system so that W is a nite Coxeter group and S is a set of simple generators of W . The length of w 2 W , l(w), is the smallest l such that w = s1s2 sl (10) where si 2 S. If v; w 2 W then we write v w if there is an s 2 S with v = ws and l(v) = l(w)+1. (It is easy to see that l(v) = l(ws) = l(w) 1, cf. Lemma 3.3.) Extending this relation by transitive closure, we obtain the weak Bruhat poset PW on W . Equivalently, this is the partial order obtained from the Cayley graph of W with respect to S by directing edges away from the identity element. The atoms of PW are just the elements of S. The 1̂ of PW is the element of maximum length, w0 = _S. If J S is any proper subset, then these elements generate a corresponding parabolic subgroup WJ which is a proper subgroup of W . So none of the elements w0(J) = _J is equal to w0 and so S = A(WP ) is independent. Thus Corollary 2.1 applies and we have proved the following result. Proposition 2.2 (Bj orner) Let (W;S) be a Coxeter system and let PW be the corresponding weak Bruhat order. Then for w 2 W we have (w) = ( ( 1)jJj if w = w0(J) for some J S, 0 else. Bjorner actually derives the Mobius function from any interval [v; w] in PW . But this follows easily from the preceding proposition since there is a poset isomorphism [v; w] [0̂; v 1w]. Next we consider the Tamari lattices [5, 7, 9]. Consider the set of all proper parenthesizations of the word x1x2 : : : xn+1. It is well known that the number of such is the Catalan number Cn = 1 n+1 2n n . Partially order this set by saying that is covered by if = : : : ((AB)C) : : : and = : : : (A(BC)) : : : for some subwords A;B;C. This poset is the Tamari lattice Tn and T3 is illustrated in Figure 2 (a). A left bracket vector, (v1; : : : ; vn), is an integer vector satisfying the conditions 6 s (x1(x2(x3x4))) s (x1((x2x3)x4)) s ((x1(x2x3))x4) s (((x1x2)x3)x4) s((x1x2)(x3x4)) HHHH JJJJJJ s (1; 2; 3) s (1; 2; 2) s (1; 2; 1) s (1; 1; 1) s(1; 1; 3) HHHH JJJJJJ (a) Parenthesized version (b) Left bracket version Figure 2: The Tamari lattice T3 1. 1 vi i for all i and 2. if Si = fvi; vi + 1; : : : ; ig then for any pair Si; Sj either one set contains the other or Si \ Sj = ;. The number of left bracket vectors with n components is also Cn. In fact given any parenthesized word we have an associated left bracket vector v( ) = (v1; : : : ; vn) de ned as follows. To calculate vi, start at xi in and move to the left, counting the number of x's and the number of left parentheses you meet until these two numbers are equal. Then vi = j where xj is the last x which is passed before the numbers balance. It is not hard to show that this gives a bijection between parenthesizations and left bracket vectors, thus inducing a partial order on the latter. This version of T3 is shown in Figure 2 (b). We will need the following theorem which is proved (in a dual version) in [9]. Theorem 2.3 (Huang and Tamari) The poset Tn is a lattice. In fact, if v( ) = (v1; : : : ; vn) and v( ) = (w1; : : : ; wn) then v( _ ) = (maxfv1; w1g; : : : ;maxfvn; wng): We can now calculate the Mobius function of the Tamari lattice. This calculation has been done before by a number of di erent people. J. M. Pallo [13] derived the result by a method equivalent to ours. Paul Edelman [private communication] demonstrated that the Mobius function is always 1 by showing that the associated order complex has the homotopy type of a wedge of spheres. Finally Bjorner and Wachs [3] used their theory of nonpure shellings to get Edelman's result. 7 Proposition 2.4 Let 2 Tn have vector v( ) = (v1; : : : ; vn). Then ( ) = ( ( 1)t if vi 2 f1; ig for all i 0 else where t is the number of vi = i 6= 1. In particular (Tn) = ( 1)n 1: Proof. Note that Tn has n 1 atoms a2; : : : ; an where v(ai) has vi = i and all other vj = 1. From Theorem 2.3 we see that the atom set is independent. Thus Corollary 2.1 applies and the given formulae follow easily. 3 Crosscuts, Euler characteristics, and characteristic polynomials We now present some proofs of related results using involutions. The following is a special case of Rota's Crosscut Theorem [14]. Theorem 3.1 (Rota) If L is a nite lattice and x 2 L then de ne ai(x) = number of sets of i atoms whose join is x. We have (x) = a0 a1 + a2 : Proof. The proof follows the same lines as that of Theorem 1.2. In this case the set is S = fA : A A(L) and _ A xg with sign function (A) = ( 1)jAj: Given any xed atom a x we de ne the involution (A) = A4 a: This is clearly well-de ned and the proof follows. It would be interesting to nd of a proof of the Crosscut Theorem in its full generality using involutions. The stumbling block is that to apply this method one would need to have a crosscut C such that for every x 2 L not covering 0̂ we have 8 C \ [0̂; x] is a crosscut of that interval. But this condition forces C to be the set of atoms. The Mobius function of any partially ordered set L can be viewed as a reduced Euler characteristic. If x 2 L then a chain of length i in the open interval (0̂; x) is c : x0 < x1 < : : : < xi where 0̂ < xj < x for all j. Let ci(x) = number of chains of length i in (0̂; x): Note that if x > 0̂ then c 1(x) = 1 because of the empty chain. Walker [16, Theorem 1.6] notes that the following result, usually known as Philip Hall's Theorem [8, 14], can be proved using involutions. Theorem 3.2 (Hall) If L is any partially ordered set with a 0̂ and x 2 L, then (x) = ( 1 if x = 0̂; c 1(x) + c0(x) c1(x) + else. (11) Proof. Again, the proof follows the lines of Theorem 1.2. Let S = f0̂g [ f(c; y) : c is a chain in (0̂; y), 0̂ < y xg with sign function (0̂) = 1 and (c; y) = ( 1)l(c) where l(c) is the length of the chain. The involution is de ned by 0̂ $ (;; x) and for (c; y) 2 S n f0̂; (;; x)g let (c; y) = ( (c n c0; c0) if y = x; (c < y; x) else. where c0 is the largest element of c, and c < y is the chain formed by adjoining y to c. The fact that this is a sign-reversing involution can now be used to show that the right side of (11) satis es the same recursion as (x). If a geometric lattice comes from a hyperplane arrangement, even more can be said about its Mobius function. Any terms in the following discussion which are not de ned can be found in the book of Orlik and Terao [11]. Let W be a nite Euclidean re ection group acting on a vector space V . Let AW be the 9 corresponding hyperplane arrangement with intersection lattice LW , i.e, LW is the set of all subspaces of V that can be obtained as intersections of hyperplanes in AW ordered by reverse inclusion. De ne the absolute length of w 2 W , l̂(w), to be the smallest l such that w can be written as w = t1t2 tl (12) with the ti coming from the set of all re ections T W . This di ers from the de nition of ordinary length given in (10) in that one is not restricted to a set S of simple re ections. An expression of the form (12) will be called absolutely reduced. We will need the following result about absolute length. Lemma 3.3 Let W be a nite re ection group and consider w 2 W . If t 2 W is any re ection then l̂(wt) = l̂(w) 1: Proof. If w = t1t2 tk is an absolutely reduced expression then wt = t1 tkt so that l̂(wt) l̂(w) + 1. Now replacing w by wt in the last inequality yields l̂(wt) l̂(w) 1. Finally, we cannot have l̂(wt) = l̂(w) since det(wt) = det(w) and det(u) = ( 1)l̂(u) for any u 2 W . For any element w 2 W let V w = fv 2 V : w(v) = vg: It follows from an easy-to-prove result of Carter [4] that if V w = X for some subspace X 2 LW then l̂(w) = codimX. This makes the statement of the following theorem unambiguous. Theorem 3.4 Let W be a nite re ection group with corresponding intersection lattice LW . Then for any X 2 LW we have (X) = ( 1)l̂(number of w 2 W with V w = X): where l̂ = l̂(w) of some (any) w with V w = X. Proof. This proof was discovered by Victor Reiner [personal communication] using the ideas in our proof of Theorem 1.2. I thank him for letting me reproduce it here. Let ~ (X) = ( 1)l̂(number of w 2 W with V w = X): Then we must show that X Y X ~ (Y ) = 0̂;X (13) 10 If X = 0̂ = V then both sides of (13) are clearly 1. If X > 0̂ then consider the setW 0 = fw 2 W : V w Xgwith sign function(w) = ( 1)l̂(w):Clearly the right side of (13) is given byPw2W 0 (w). But W 0 is just the stabilizerof X, and so is a non-trivial re ection group in its own right. Let t be any xedre ection in W 0 and de ne an involution : W 0 ! W 0 by(w) = wt:By Lemma 3.3 this is sign-reversing and so we are done.We should note that there is a direct connection between absolutely reducedexpressions and NBC bases. Speci cally, in [1] Barcelo, Goupil and Garsia showthat if H1; : : : ; Hm is an NBC base of AW then the corresponding product ofre ections rH1 rHm is totally reduced and this gives a bijection between NBCbases and W .We end by showing how Theorem 3.4 relates the characteristic polynomial ofLW to the Poincare polynomial of W . The characteristic polynomial of LW is thegenerating function for its Mobius function:(LW ; t) = XX2LW (X)tdimX :The Poincar e polynomial of W is the generating function for its elements by ab-solute length:(W; t) = Xw2W tl̂(w):Theorem 3.5 Let W be a nite re ection group in V , dimV = n, with corre-sponding intersection lattice LW . Then(W; t) = ( t)n (LW ; 1=t):Proof. Using Theorem 3.4 and the lemma of Carter cited previously, we have thefollowing series of equalities( t)n (LW ; 1=t) = XX2LW (X)( t)codimX= Xw2W tl̂(w)= (W; t):11 Acknowledgements. I would like to thank Victor Reiner for many helpful dis-cussions and in particular for posing the problem of nding an involution proof ofthe Crosscut Theorem. Anders Bjorner suggested considering the lattices discussedin Section 2 as applications. Finally, I would like to thank Christian Krattenthalerand Gian-Carlo Meloni who independently asked for a combinatorial explanationof the arbitrariness of the atom ordering in the geometric case.References[1] H. Barcelo and A. Goupil, Non broken circuits of re ection groups and theirfactorization in Dn, preprint.[2] A. Bjorner, Orderings of Coxeter groups, Contemporary Math. 34 (1984),175{195.[3] A. Bjorner and M. Wachs, Shellable nonpure complexes and posets, in prepa-ration.[4] R. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972),1{59.[5] H. Friedman and D. Tamari, Problemes d'associativite: Une treillis nis in-duite par une loi demi-associative, J. Combin. Theory 2 (1967), 215{242.[6] A. M. Garsia and S. C. Milne, A Rogers-Ramanujan bijection, J. Combin.Theory Ser. A bf 31 (1981), 289{339.[7] G. Gratzer, \Lattice Theory," Freeman and Co., San Francisco, CA, 1971, pp.17{18, problems 26-36.[8] P. 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