Toric Fiber Products Seth Sullivant North Carolina State University June 8, 2011 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 1 / 26
Families of Ideals Parametrized by Graphs Let G be a finite graph Let R G a polynomial ring associated to G Let I G R G an ideal associated to G Problem Classify the graphs G such that I G satisfies some nice property. Often I G := ker φ G for some ring homomorphism φ G. Problem Determine generators for I G. How do they depend on the graph G? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 2 / 26
Graph Decompositions Question Are decompositions of the graphs G = G 1 #G 2 reflected in the ideals I G = I G1 #I G2? How to study ideals of large graphs by breaking into simple pieces? + = + = Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 3 / 26
Toric Fiber Product Let A = {a 1,..., a r } Z d. Let K[x] := K[x i j : i [r], j [s]], with deg x i j = a i. Let K[y] := K[y i k : i [r], k [t]], with deg y i k = ai. Let K[z] := K[z i jk : i [r], j [s], k [t]], with deg zi jk = ai. Let φ : K[z] K[x] K K[y] = K[x, y] defined by z i jk x i j y i k for all i, j, k. Definition Let I K[x], J K[y] ideals homogeneous w.r.t. grading by A. The toric fiber product of I and J is the ideal I A J = φ 1 (I + J). Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 4 / 26
Two Important Examples Example (Coarse Grading) Let K[x], K[y] have common grading deg x i = deg y j = 1 for all i, j. Then I K[x], J K[y] homogeneous, are homogeneous in the standard/coarse grading. I A J K[z] is the ordinary Segre product ideal. Example (Fine Grading) Let K[x 1,..., x r ], K[y 1,..., y r ] have common grading deg x i = deg y i = e i for all i. Then I K[x], J K[y] homogeneous, are monomial ideals. I A J = I(z) + J(z) K[z 1,..., z r ] is the sum of monomial ideals. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 5 / 26
A More Complex Example Let I = x kl1 m 1 x kl2 m 2 x kl1 m 2 x kl2 m 1 : k [r 1 ], l 1, l 2 [r 2 ], m 1, m 2 [r 3 ] Let J = y l1 m 1 ny l2 m 2 n y l1 m 2 ny l2 m 1 n :, l 1, l 2 [r 2 ], m 1, m 2 [r 3 ], n [r 4 ] Let deg x klm = deg y lmn = e l e m Define φ : K[z klmn : k [r 1 ],...] K[a kl, b km, c ln, d mn : k [r 1 ],...] by z klmn a kl b km c ln d mn Then I A J = ker φ. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 6 / 26
Where s the Fiber Product? Suppose I, J are toric ideal I = I B, J = I C, B Z e 1, C Z e 2 B = {b i j : i [r], j [s]} C = {c i k : i [r], k [t]} If I B homogeneous with respect to A, then there is a linear map π 1 : Z e 1 Z d, π 1 (b i j ) = ai. If I C homogeneous with respect to A, then there is a linear map π 2 : Z e 2 Z d, π 2 (c i k ) = ai. Then I A J is a toric ideal, whose vector configuration is a fiber product B A C. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 7 / 26
Codimension-0 Toric Fiber Products Definition The codimension of a TFP is the codimension of the toric ideal I A. I A J has codimension 0 iff A is linearly independent. Proposition Suppose A linearly independent. Let If deg m = deg m then n = n and m = x i 1 j1 x in j n and m = x i 1 j 1 x i n j n. i 1 = i 1,..., i n = i n. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 8 / 26
Persistence of Normality and Generating Sets Theorem (Ohsugi, Michałlek, etc.) Let I K[x], and J K[y] be toric ideals, and A linearly independent. Then K[z]/(I A J) normal K[x]/I and K[y]/J normal. If f K[x], homogeneous w.r.t. A, write f = c u x i 1 ju1 x in j un. Lift to c u z i 1 ju1 k 1 z in j unk n. Theorem Let A linearly independent. Then I A J generated by 1 Lifts of generators of I and J 2 Obvious quadrics z i j 1 k 1 z i j 2 k 2 z i j 1 k 2 z i j 2 k 1 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 9 / 26
GIT Quotients NA = {λ 1 a 1 +... + λ r a r : λ i N} R = K[x]/I is an NA graded ring. R = a NA R a S = K[y]/J is an NA graded ring. S = a NA S a Proposition Suppose A linearly independent. Then If K = K, then K[z]/(I A J) = a NA R a K S a. Spec(K[z]/(I A J)) = (Spec(K[x]/I) Spec(K[y]/J)) //T, where T acts on Spec(K[x]/I) Spec(K[y]/J) via t (x, y) = (t x, t 1 y). Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 10 / 26
Higher Codimension Toric Fiber Products Not a GIT quotient No hope for general construction of generators When is normality preserved? Proposition Let K = K. Suppose that I = i P i, and J = j Q j are primary decompositions. Then I A J = i,j (P i A Q j ) is a primary decomposition of I A J. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 11 / 26
Generators of Higher Codim TFPs Suppose I K[x], J K[y], are toric ideals. A = {a 1,..., a r }, Let K[w] := K[w 1,..., w r ]. Let ψ xw : K[x] K[w], x i j w i (similarly ψ yw ) Definition Let Ĩ = x u x v I : φ(x u x v ) = 0 J = y u y v J : φ(y u y v ) = 0 The ideal Ĩ Ã J is the associated codimension 0 TFP. Ĩ Ã J I A J Ĩ Ã J is usually related (via graph theory) in a nice way to I A J. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 12 / 26
Gluing Generators Let and f = x i 1 j1 x in j n x i 1 j x i n 1 j I n g = y i 1 k1 y in k n y i 1 k 1 y i n k n J that is, φ xw (f ) = φ yw (g). Then glue(f, g) = z i 1 j1 k 1 z in j nk n z i 1 j z i 1 k 1 n j n k I n A J Question Two natural classes of generators of I A J: when do they suffice? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 13 / 26
Awesome Pictures that Explain Everything U Answer Gluing and the associated codim 0 tfp always suffice to generate I A J. But... how to find the right binomials to glue? Projecting a fiber onto ker Z A. 0 2 3 0 1 3 = 0 3 Projected and connected fibers need not be compatible Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 14 / 26
Summary of General Results on Toric Fiber Products Codim 0 TFPs Can Explicitly Describe Generators/ Gröbner bases from I and J Normality Preserved for Toric Ideals Geometric Interpretation as GIT Quotient Arbitrary Codim TFPs Primary Decompositions Multiply Can Explicitly Describe Generators given generators of I and J with special properties (toric case only) Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 15 / 26
Markov Bases Definition Let A : Z n Z d be a linear transformation. A Markov Basis for A is a finite subset B ker Z (A) such that for all u, v N n with A(u) = A(v) there is a sequence b 1,..., b L B such that 1 u = v + L i=1 b i, and 2 v + l i=1 b i 0 for l = 1,..., L. Markov bases allow us to take random walks over the set of nonnegative integral points inside of polyhedra. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 16 / 26
Example: 2-way tables Let A : Z k 1 k 2 Z k 1+k 2 such that m m A(u) = u 1j,..., u k1 j; j=1 j=1 k u i1,..., i=1 k i=1 = vector of row and column sums of u u ik2 ker Z (A) = {u Z k 1 k 2 : row and columns sums of u are 0} Markov basis consists of the 2 ( )( k 1 k2 ) 2 2 moves like: 0 0 0 0 1 0 1 0 1 0 1 0 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 17 / 26
Fundamental Theorem of Markov Bases Definition Let A : Z n Z d. The toric ideal I A is the ideal p u p v : u, v N n, Au = Av K[p 1,..., p n ], where p u = p u 1 1 pu 2 2 pun n. Theorem (Diaconis-Sturmfels 1998) The set of moves B ker Z A is a Markov basis for A if and only if the set of binomials {p b+ p b : b B} generates I A. 0 0 0 0 1 0 1 0 p 21 p 33 p 23 p 31 1 0 1 0 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 18 / 26
More Complicated Marginals Let Γ = {F 1,..., F r }, each F i {1, 2,..., n}. Let d = (d 1,..., d n ) and u Z d 1 d n 0. Let A Γ,d (u) = (u F1,..., u Fr ), lower order marginals. 1 2 2-way margins of 4-way table: {1, 2}, {2, 3}, {3, 4}, {1, 4} -margins Question 4 3 How does the Markov basis of A Γ,d depend on Γ and d? How do the generators of I Γ,d depend on Γ and d? Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 19 / 26
Decomposing Along Face of Simplicial Complex Theorem (Dobra-Sullivant 2004, Hoşten-Sullivant 2002) Suppose that Γ = Γ 1 Γ 2, and Γ 1 Γ 2 is a face of both. Then and A is linearly independent. I Γ,d = I Γ1,d 1 A I Γ2,d 2 + = These complexes are called reducible in statistics. Allows for direct construction of Markov bases of reducible models. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 20 / 26
Decomposing Along Arbitrary Subcomplexes Theorem Suppose that Γ = Γ 1 Γ 2, and Γ 1 Γ 2 =. Then I Γ,d = I Γ1,d 1 A,d I Γ2,d 2, and, associated codim zero TFP is I Γ 2,d. + = If overlap is small, and associated codim zero TFP is simple, can construct generators of I Γ,d. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 21 / 26
Example Applications Theorem (Kral, Norin, Pragnac 2010) + = If d i = 2 for all i, then I Γ,d is generated in degree 4 if Γ is a series-parallel graph (no K 4 -minors). Theorem (4-cycle) 1 2 If d 2 = d 4 = 2, then I Γ,d generated in degree 2 and 4, for all d 1, d 3. 4 3 Theorem If Γ is the boundary of a bipyramid over a simplex of dimension d, and all d i = 2, I Γ,d generated in degrees 2 d+1, 2 d, 2. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 22 / 26
Conditional Independence Ideals X = (X 1,..., X n ) is an n-dimensional discrete random vector. Probability Distribution Let (A, B, C), partition of [n]. p i1 i n = Prob(X 1 = i 1,..., X n = i n ) Conditional independence statement X A X B X C gives a binomial ideal in C[p i1 i n : i j [r j ]]. Example Let n = 4, r 1 = r 2 = r 3 = r 4 = 2, X 1 X 3 (X 2, X 4 ), gives CI ideal I X1 X 3 (X 2,X 4 ) = p 1111 p 2121 p 1121 p 2111, p 1112 p 2122 p 1122 p 2112, p 1211 p 2221 p 1221 p 2211, p 1212 p 2222 p 1222 p 2212 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 23 / 26
Global Conditional Independence Ideals G undirected graph, vertex set [n] CI statement X A X B X C holds for G if C separates A from B in G. Let Global(G) set of all CI statements holding for G. I Global(G) = X A X B X C Global(G) I XA X B X C Example (4-cycle) 1 2 I Global(G) = I X1 X 3 (X 2,X 4 ) + I X 2 X 4 (X 1,X 3 ) 4 3 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 24 / 26
Clique Sums Proposition If G = G 1 #G 2 is a clique sum, then and A is linearly independent. Example I Global(G) = I Global(G1 ) A I Global(G2 ) + = For r i = 2 for all i, I Global(G) has 9 9 = 81 prime components. Conjecture For all graphs G, I Global(G) is a radical ideal. All nontoric components are related to marginal positivity. Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 25 / 26
Summary Problems in Algebraic Statistics call for the study of ideals associated to graphs and simplicial complexes Graph theory provides decomposition theory Algebraic structure of ideals reflected in structure of graphs (conjecturally) Toric fiber product is an algebraic decomposition tool for proving results for some graph classes References A. Engström and T. Kahle. Multigraded commutative algebra of graph decompositions. 1102.2601 S. Sullivant. Toric fiber products. J. Algebra 316 (2007), no. 2, 560 577. math.ac/0602052 Seth Sullivant (NCSU) Toric Fiber Products June 8, 2011 26 / 26