Monomial Bases for NBC Complexes Jason I. Brown Department of - PDF document

Monomial Bases for NBC Complexes Jason I. Brown Department of Mathematics and Statistics Dalhousie University Halifax, NS B3H 3J5, CANADA Bruce E. Sagan Department of Mathematics Michigan State University East Lansing, MI 48824-1027, USA sagan@math.msu.edu www.math.msu.edu/˜sagan 1. Complexes and chromatic polynomials 2. NBC complexes 3. Stanley-Reisner rings and hsop’s 4. Monomial ideals 5. Comments 1

1. Complexes and chromatic polynomials Let ∆ be a simplicial complex on a finite set E , so ∆ is a family of subsets of E satisfying S ∈ ∆ and T ⊆ S implies T ∈ ∆. The S ∈ ∆ are called faces . We assume ∆ is pure of rank r meaning that | S | = r for all maximal faces S ∈ ∆. For 0 ≤ i ≤ r , let f i = f i (∆) = # of faces S ∈ ∆ with | S | = i. The f -polynomial of ∆ is f ( x ) = f 0 + f 1 x + f 2 x 2 + · · · + f r x r . The h -polynomial of ∆ is x � � (1 − x ) r f h ( x ) = 1 − x f 0 (1 − x ) r + f 1 x (1 − x ) r − 1 + · · · + f r x r . = and let h i = h i (∆) = coefficient of x i in h ( x ). 2

Let G = ( V, E ) be a graph with | V | = p and | E | = q . A proper coloring of G is c : V → { 1 , 2 , . . . , λ } such that vw ∈ E implies c ( v ) � = c ( w ) . The chromatic polynomial of G is P ( G ) = P ( G ; λ ) = # of such proper colorings. v Example. Let ④ � ❅ � ❅ � ❅ G = w � ❅ ④ u ④ ❅ � ❅ � ❅ � ❅ � ④ t P ( G ) = # ways to color t , then u , then v , then w = λ ( λ − 1)( λ − 2)( λ − 2) λ 4 − 5 λ 3 + 8 λ 2 − 4 λ. = Proposition 1 Let K p be the edgeless graph and let T be a tree on p vertices. Then P ( T ; λ ) = λ ( λ − 1) p − 1 . P ( K p ; λ ) = λ p and 3

Let G be a graph and e ∈ E . Let G \ e = G with e deleted, G/e = G with e contracted. v ① e � ❅ Example. Let � ❅ G = w � ❅ ① ① ❅ � ❅ � ❅ � ① v v = w ① ① ❅ ❅ ❅ ❅ G \ e = G/e = w ❅ ❅ ① ① ① ❅ � � ❅ � � ❅ � � ① ① Theorem 2 (Deletion-Contraction) For e ∈ E P ( G ) = P ( G \ e ) − P ( G/e ) Proof. If e = vw then P ( G \ e ) = (# proper c for G \ e s.t. c ( v ) � = c ( w )) +(# proper c for G \ e s.t. c ( v ) = c ( w )) = P ( G ) + P ( G/e ) . Corollary 3 For any graph G : 1. P ( G ; λ ) is a monic polynomial in λ . 2. deg P ( G ; λ ) = p = | V | . 3. Coefficients of P ( G ; λ ) alternate in sign. 4

2. NBC complexes Define coefficients f i by P ( G ; λ ) = f 0 λ p − f 1 λ p − 1 + · · · and coefficients h i by P ( G ; λ ) = h 0 λ ( λ − 1) p − 1 − h 1 λ ( λ − 1) p − 2 + · · · . Let C = C ( G ) = set of cycles/circuits of G . Let G be ordered meaning that E has been given a total order e 1 < e 2 < . . . < e q . Then each C ∈ C has broken circuit C = C − min C. The NBC complex of G is ∆ = ∆( G ) = { S ⊆ E : S contains no C } . Then ∆( G ) is a pure simplicial complex. Theorem 4 Let P ( G ; λ ) have coefficients f i and h i as defined above. Then for 0 ≤ i ≤ p f i = f i (∆( G )) and h i = h i (∆( G )) . 5

❅ 4 ④ � 2 � ❅ � ❅ G = � ❅ 3 ④ ④ ❅ � ❅ � 5 ❅ � 1 ❅ � ④ ④ ④ ④ ❅ 4 ❅ 4 � � 2 2 � ❅ � ❅ � ❅ � ❅ C ( G ) = � ❅ � ❅ 3 3 ④ ④ ④ ④ ❅ � ❅ � ❅ � ❅ � 5 ❅ � 1 5 ❅ � 1 ❅ � ❅ � ④ ④ ④ ¯ C ( G ) = { 35 , 34 , 245 } ∆( G ) = {∅} ∪ { 1 , 2 , 3 , 4 , 5 } ∪{ 12 , 13 , 14 , 15 , 23 , 24 , 25 , 45 } ∪{ 123 , 124 , 125 , 145 } ( f i (∆)) = (1 , 5 , 8 , 4 , 0) . P ( G ; λ ) = λ ( λ − 1)( λ − 2) 2 = λ 4 − 5 λ 3 + 8 λ 2 − 4 λ. � � � � � � ❅ ; λ t ❅ ; λ t t ❅ ; λ � ❅ � ❅ ❅ P = P − P t t t t t ❅ � � ❅ � � � ❅ ❅ � t t t � � � � � � � � t t t t = P � ; λ − P ❅ ; λ − P � ; λ + P ; λ t t t t ❅ � ❅ ❅ � t t t t = λ ( λ − 1) 3 − λ ( λ − 1) 2 − λ ( λ − 1) 2 + λ ( λ − 1) = λ ( λ − 1) 3 − 2 λ ( λ − 1) 2 + λ ( λ − 1) ( h i (∆)) = (1 , 2 , 1 , 0 , 0) 6

3. Stanley-Reisner rings and hsop’s Let F [ x ] be the polynomial ring over field F with variables x = { x 1 , . . . , x q } . If E = { e 1 , . . . , e q } then S ⊆ E has monomial x S = � x i . e i ∈ S Simplicial complex ∆ has Stanley-Reisner ring F (∆) = F [ x ] / ( x S : S �∈ ∆) . In particular, for an ordered graph G we let F ( G ) = F (∆( G )) = F [ x ] / ( x C : C ∈ C ( G )) . Now F ( G ) has a homogeneous system of parame- ters (hsop) of degree one θ 1 , . . . , θ t , i.e., 1. θ i is linear without constant term for all i , 2. θ 1 , . . . , θ t are algebraically independent, 3. F ( G ) / ( θ 1 , . . . , θ t ) is finite dim. over F . Brown gave an explicit hsop for F ( G ). WLOG G is connected and let T be a spanning tree of G . If e ∈ E ( T ) then e has fundamental disconnecting set D e = D e ( G ) = { f ∈ E ( G ) : T − e + f connected } and hsop element (when F = Z 2 ) � θ e = x i . e i ∈ D e 7

Example 7 7 ✉ ✉ ✉ ✉ � ❅ ❅ 1 6 6 � ❅ ❅ � ❅ ❅ � ❅ ❅ G = T = ✉ ✉ ✉ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ ◗ ✑✑✑✑✑✑ 2 5 5 ◗ ◗ ◗ ◗ ◗ ◗ 3 ◗ 4 3 ◗ 4 ◗ ◗ ✉ ✉ C ( G ) = { 13467 , 2345 , 12567 } Z 2 ( G ) = Z 2 [ x 1 , . . . , x 7 ] / ( x 3 x 4 x 6 x 7 , x 3 x 4 x 5 , x 2 x 5 x 6 x 7 ) 7 ✉ ✉ � ❅ 1 6 � ❅ � ❅ θ 3 = x 3 + x 1 + x 2 � ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 2 5 ◗ ◗ ◗ 3 ◗ 4 ◗ ✉ 7 ✉ ✉ � ❅ 1 6 � ❅ � ❅ θ 4 = x 4 + x 1 + x 2 � ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 2 5 ◗ ◗ ◗ 3 4 ◗ ◗ ✉ 7 ✉ ✉ ❅ 6 ❅ ❅ θ 5 = x 5 + x 2 ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 2 5 ◗ ◗ ◗ 3 ◗ 4 ◗ ✉ 7 ✉ ✉ � ❅ 1 6 � ❅ � ❅ θ 6 = x 6 + x 1 � ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 5 ◗ ◗ ◗ 3 ◗ 4 ◗ ✉ 7 ✉ ✉ � ❅ 1 6 � ❅ � ❅ θ 7 = x 7 + x 1 � ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 5 ◗ ◗ ◗ 3 ◗ 4 ◗ ✉ 8

4. Monomial ideals If F (∆) has an hsop θ 1 , . . . , θ t we let R (∆) = F (∆) / ( θ 1 , . . . , θ t ) . Consider Mon( k ) = set of monomials in F [ x 1 , . . . , x k ]. A subset L ⊆ Mon( k ) is a lower order ideal if m ∈ L and n | m imples n ∈ L. The lower order ideal generated by S ⊆ Mon( k ) is L ( S ) = { m ∈ Mon( k ) : m | n for some n ∈ S } . Upper ideal and U ( S ) are defined dually. Theorem 5 (Macaulay, Stanley) Suppose ∆ is a simplicial complex and that the ring F (∆) is Cohen- Macaulay. Then R (∆) has a basis, L , which is a lower order ideal of monomials and h i (∆) = # of monomials of total degree i in L . 9

For a graph G , F ( G ) is Cohen-Macaulay. We have a conjectured construction of a basis for R ( G ). An ordering e 1 < . . . < e q is standard if the last p − 1 edges form a tree. Let k = | E ( G ) − E ( T ) | . We can pick the monomial basis for R ( G ) inside Mon( k ) since Brown’s θ i can be used to eliminate the other variables, replacing each x C by a polynomial p C . Example. In our running example, k = 2 and Z 2 ( G ) = Z 2 [ x 1 , . . . , x 7 ] / ( x 3 x 4 x 6 x 7 , x 3 x 4 x 5 , x 2 x 5 x 6 x 7 ) . θ 3 = x 3 + x 1 + x 2 , θ 4 = x 4 + x 1 + x 2 , θ 5 = x 5 + x 2 . So, picking one of the broken circuit monomials x C = x 3 x 4 x 5 p C = ( x 1 + x 2 ) 2 x 2 . becomes For 1 ≤ i ≤ k , the graph T + e i has a unique fun- damental circuit C i . Conjecture 6 Let G be connected. Then there is a standard ordering of E such that R ( G ) has basis L ( G ) = Mon( k ) − U ( m C : C ∈ C ( G )) where  x # C i if C = C i fundamental,  m C = i min p C else.  Here min p picks out the lexicographically smallest monomial in p . 10

7 7 ① ① ① ① � ❅ ❅ 1 6 6 � ❅ ❅ � ❅ ❅ � ❅ ❅ � ❅ ❅ � ❅ ❅ G = T = ① ① ① ① ① ① ◗ ✑ ◗ ✑ 2 5 5 ◗ ✑✑✑✑✑✑✑✑ ◗ ✑✑✑✑✑✑✑✑ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ 3 4 3 4 ◗ ◗ ◗ ◗ ① ① Fundamental cycles: 7 ✉ ✉ � ❅ 1 6 � ❅ � ❅ m C 1 = x 4 C 1 = { 1 , 3 , 4 , 6 , 7 } � ❅ ✉ ✉ ✉ 1 ◗ ✑✑✑✑✑✑ 5 ◗ ◗ ◗ 3 ◗ 4 ◗ ✉ 7 ✉ ✉ ❅ 6 ❅ ❅ m C 2 = x 3 C 2 = { 2 , 3 , 4 , 5 } ❅ ✉ ✉ ✉ 2 ◗ ✑✑✑✑✑✑ 2 5 ◗ ◗ ◗ 3 4 ◗ ◗ ✉ Nonfundamental cycle: 7 m C 3 = x 2 1 x 2 C 3 = { 1 , 2 , 5 , 6 , 7 } ✉ ✉ � ❅ 2 1 6 � ❅ � ❅ x C 3 = x 2 x 5 x 6 x 7 � ❅ ✉ ✉ ✉ ◗ ✑✑✑✑✑✑ 2 5 ◗ ◗ ◗ p C 3 = x 2 x 2 x 1 x 1 3 ◗ 4 ◗ ✉ So R ( G ) has basis L ( G ) = Mon(2) − U ( x 4 1 , x 3 2 , x 2 1 x 2 2 ) . 11