Properties of ideals in an invariant ascending chain

Transcription

1 Properties of ideals in an invariant ascending chain Uwe Nagel (University of Kentucky) joint work with Tim Römer (University of Osnabrück) Daejeon, August 3, 2015

2 Two-way Tables Fix integers c, r 2 and consider the map ϕ r : Z c r Z c Z r that computes the row and column sums of a c r matrix. A Markov basis of ker ϕ r is B r = {e i1,j 1 +e i2,j 2 e i1,j 2 e i2,j 1 i 1, i 2 [c] = {1, 2,..., c}, j 1, j 2 [r]} For c = 3, r = 4, a typical element is The toric ideal corresponding to B r is I r = (x i1,j 1 x i2,j 2 x i1,j 2 x i2,j 1 ) K [X [c] [r] ] x 1,1 x 2,4 x 1,4 x 2,

3 Two-way Tables Note ( ) x1,1 x x 1,1 x 2,4 x 1,4 x 2,1 = det 1,4 x 2,1 x 2,4 Thus, I r is generated by the 2-minors of a generic c r matrix x 1,1 x 1,r X c r =., i.e., x c,1... x c,r I r = I 2 (X c r ).

4 Two-way Tables Vary r, and consider in the polynomial ring K [X [c] N] ] = K [x i,j i [c], j N} the ideal I r K [X [c] N] ] = I 2 (X c r ) K [X [c] N] ] = I 2 (X c ). r 2 r 2 It is not f.g. However, up to permutations of rows and columns there is only one move/generator, namely x 1,1 x 2,2 x 1,2 x 2,1.

5 Two-way Tables Vary r, and consider in the polynomial ring K [X [c] N] ] = K [x i,j i [c], j N} the ideal I r K [X [c] N] ] = I 2 (X c r ) K [X [c] N] ] = I 2 (X c ). r 2 r 2 It is not f.g. However, up to permutations of rows and columns there is only one move/generator, namely x 1,1 x 2,2 x 1,2 x 2,1. Consider only permutations of columns, so π Sym(r) acts on K [X [c] [r] ] by π x i,j = x i,π(j). Induces an action of Sym( ) = r N Sym(r) on K [X [c] N]].

6 Two-way Tables Vary r, and consider in the polynomial ring K [X [c] N] ] = K [x i,j i [c], j N} the ideal I r K [X [c] N] ] = I 2 (X c r ) K [X [c] N] ] = I 2 (X c ). r 2 r 2 It is not f.g. However, up to permutations of rows and columns there is only one move/generator, namely x 1,1 x 2,2 x 1,2 x 2,1. Consider only permutations of columns, so π Sym(r) acts on K [X [c] [r] ] by π x i,j = x i,π(j). Induces an action of Sym( ) = r N Sym(r) on K [X [c] N]]. Observe: (i) I = I 2 (X c ) is invariant under the action of Sym( ). (ii) I is generated by finitely many Sym( )-orbits.

7 Sym( )-invariant Ideals Problem Study Sym( )-invariant ideals.

8 Sym( )-invariant Ideals Problem Study Sym( )-invariant ideals. Frameworks: (i) Hillar, Sullivant (ii) Draisma (and Kuttler, Eggermont, Krone, Leykin) (iii) Sam, Snowden

9 Sym( )-invariant Ideals Problem Study Sym( )-invariant ideals. Frameworks: (i) Hillar, Sullivant (ii) Draisma (and Kuttler, Eggermont, Krone, Leykin) (iii) Sam, Snowden Theorem (Hillar, Sullivant, 2012) If I K [X [c] N] ] is Sym( )-invariant, then I is generated by finitely many Sym( )-orbits.

10 Inc-invariant Ideals Consider the set of (strictly) increasing functions on N Inc = Inc(N) = {π : N N π(i) < π(i + 1) for all i 1}. It is a monoid. (If π, σ Inc, then π σ Inc.) It acts on K [X [c] N ] by π x i,j = x i,π(j). Note Inc Sym( ).

11 Inc-invariant Ideals Consider the set of (strictly) increasing functions on N Inc = Inc(N) = {π : N N π(i) < π(i + 1) for all i 1}. It is a monoid. (If π, σ Inc, then π σ Inc.) It acts on K [X [c] N ] by π x i,j = x i,π(j). Note Inc Sym( ). Lemma For each polynomial f K [X [c] N ], one has Inc f Sym( ) f.

12 Inc-invariant Ideals Consider the set of (strictly) increasing functions on N Inc = Inc(N) = {π : N N π(i) < π(i + 1) for all i 1}. It is a monoid. (If π, σ Inc, then π σ Inc.) It acts on K [X [c] N ] by π x i,j = x i,π(j). Note Inc Sym( ). Lemma For each polynomial f K [X [c] N ], one has Inc f Sym( ) f. Proof. Let π Inc, and let k be the largest column index of a variable appearing in f. Choose some σ Sym( ) satisfying { π(i) if i k σ(i) = i if i > π(k). Then π f = σ f.

13 Inc-invariant Ideals Corollary If an ideal is Sym( )-invariant, then it is Inc-invariant. Theorem (Hillar, Sullivant, 2012) If I K [X [c] N] ] is an Inc-invariant ideal, then it has a finite Inc-Gröbner basis.

14 Filtrations Simplify notation: Fix c, and set K [X r ] = K [X [c] [r] ] and K [X] = K [X [c] N ]. For integers 1 m n, define subsets of Inc Inc m,n = {π Inc π(m) n}. Example For c = 1, consider I = (x 1 x 4 ) K [X]. Then Inc 4,5 I = (x 1 x 4, x 1 x 5, x 2 x 5 ). Definition An Inc-invariant filtration I = (I r ) r N is a family of ideals I r K [X r ] such that Inc m,n I m I n whenever m n. Thus, I 1 K [X] I 2 K [X] I 3 K [X]

15 Filtrations Lemma and Definition Given an ideal 0 I r K [X r ], define its Inc-closure as (Inc I r )K [X]. Alternately, define a sequence of ideals I = (I n ) n N by { (0) if 1 n < r I n = (Inc r,n I r )K [X n ] if n r. Then I is an Inc-invariant filtration, and (Inc I r )K [X] = n N I n K [X].

16 Filtrations Let I K [X] be an Inc-invariant ideal. Define I n = I K [X n ]. Then (I n ) n N is an Inc-invariant filtration I 1 K [X] I 2 K [X] I 3 K [X] that stabilizes, that is, there is an integer r such that (Inc m,n I m ) = I n whenever n m r. Thus, I is the Inc-closure of I r. Example For c = 2, consider I = (x 1,i x 2,j i, j N). Then I is the Inc-closure of (x 1,1 x 2,1, x 1,1 x 2,2, x 1,2 x 2,1 ).

17 Filtrations Corollary Every Inc-invariant filtration I = (I n ) n N stabilizes. Proof. I = n N I nk [X] is Inc-invariant.

18 Filtrations Corollary Every Inc-invariant filtration I = (I n ) n N stabilizes. Proof. I = n N I nk [X] is Inc-invariant. Approach: Capture numerical properties of an Inc-invariant ideal by studying the ideals in such a chain simultaneously.

19 Hilbert Series M graded K [X n ]-module Hilbert function h M : Z Z, h M (j) = dim K [M] j. Hilbert series is a formal power series H M (t) = h M (j) t j j Z Hilbert: H M (t) is rational and can be written as H M (t) = f (t) (1 t) d, where f Z[t], d = dim M and 0 < f (1) = deg M is the degree or multiplicity of M.

20 Bigraded Hilbert Series Consider an Inc-invariant filtration I = (I n ) n N of homogeneous ideals I 1 K [X] I 2 K [X] I 3 K [X] Definition Abusing notation slightly, define its bigraded Hilbert series as H I (s, t) = H K [Xn]/In (t) s n = dim K [K [X n ]/I n ] j s n t j. n 1 n 1, j 0 Theorem If I = (I n ) n N is an Inc-invariant filtration of homogeneous ideals, then its bigraded Hilbert series is a rational function in s and t whose denominator is a product of polynomials, each one is of the form ((1 t) c s f (t)), where f (t) Z[t].

21 Bigraded Hilbert Series Corollary Each Inc-invariant ideal I of K [X] admits a rational Hilbert series H K [X]/I (s, t) = dim K [K [X n ]/I n ] j s n t j, where I n = I K [X n ]. Note I = n 1 I n K [X]. n 1, j 0

22 Bigraded Hilbert Series Example Every homogeneous ideal J in a polynomial ring S = K [Y 1,..., Y c ] in c variables gives rise to an Inc-invariant filtration. Indeed, let φ : S K [X 1 ] be the ring isomorphism, defined by φ(y i ) = X i,1, and define a sequence of ideals I = (I n ) n N by I n = { φ(j) if n = 1 (Inc 1,n I 1 ) if n > 1. Then I is an Inc-invariant chain, and I = n N I n is the Inc-closure of I 1 = φ(j). Its bigraded Hilbert series is where H S/J) (t) = H K [X]/I (s, t) = H I (s, t) = s f (t) (1 t) d s f (t), f (t) (1 t) d is the Hilbert series of S/J with f (1) > 0.

23 Bigraded Hilbert Series Example Fix an integer c 2, and consider a sequence of ideals I = (I n ) n N, defined by { (0) if 1 n < c I n = I 2 (X c n ) if n c. (cf. 2-way tables). The sequence I is Inc-invariant. The bigraded Hilbert series is H K [X]/I2 (X c )(s, t) = H I (s, t) (1 t) c2 s c = (1 t) c2 c [(1 t) c s] + s c (1 t) 2c 2 [1 t s] c 1 ( )( c 1 n 1 j j j=0 ) t j.