Diagonal Invariants of the symmetric group and products of linear forms

Transcription

1 Diagonal Invariants of the symmetric group and products of linear forms CRM, May 29, 2007 Emmanuel Briand. Universidad de Sevilla

2 Products of linear forms= Decomposable forms K: algebraically closed field. Decomposable forms of degree n in K[t 0, t 1,..., t r ] = totally factorizable forms = products of n linear forms. They are a closed algebraic subvariety in the space of forms of degree n. What is its ideal?

3 Two nice works on the problem John Dalbec, Multisymmetric functions, Beiträge zur Algebra und Geometrie 40 (1999). Friedrich Junker, Über symmetrische Functionen von mehreren Reihen vor Veränderlichen, Matematische Annalen 43 (1893).

4 Google translation (up to permutation)

5 Defining maps for the subvariety of decomposable forms V : K vector space of dimension r + 1. The subvariety of decomposable forms is the image of: π : (V ) n /S n S n V (l 1, l 2,..., l n ) mod S n l 1 l 2 l n It is the affine cone over the image of: Pπ : Symm n P r = (PV ) n /S n P(S n V ) (H 1, H 2,..., H n ) mod S n H 1 + H H n The map Pπ is injective and its image, Chow(n,0, P r ), is the Chow variety of the 0-cycles of degree n in P r.

6 Associated maps of graded algebra S n V = symmetric power over V (quotient of n V ). T n symv = symmetric tensors over V (subspace of n V ). The map Pπ : Symm n P r = (PV ) n /S n P(S n V ) gives rise to a map between the homogeneous coordinate rings: π : d=0 T n syms d V d=0 S d T n symv = S T n symv

7 Questions Pπ : Symm n P r P(S n V ) (H 1, H 2,..., H n ) mod S n H 1 + H H n Pb 1. Is Pπ an isomorphism Symm n P r = Chow(n,0, P r )? Pb 2. Compute, or describe, ker π (defining equations for the subvariety of decomposable forms)

8 In coordinates The map: Pπ : Symm n P r = (PV ) n /S n P(S n V ) can be worked out in homogeneous coordinates. Write l i = a i0 t 0 + a i1 t a ir t r and i l i = α ê α t α 0 0 tα 1 1 tα r r. It appears as: Â = a 10 a 11 a 20. a n0 a 1r.. a nr ( ê α (Â) ) α =n The algebra d TsymSV n of Symm n P r is HDSym r+1 n (K) (homogeneous diagonal invariants of the symmetric group): the polynomials in the entries of Â, invariants under row permutations, homogeneous in the variables of each row. The ê α (fundamental homogeneous invariants) are a linear basis for the piece of degree 1 of HDSym r+1 n (K).

9 In coordinates, locally The map: Pπ : Symm n P r = (PV ) n /S n P(S n V ) can be worked out in affine charts: a i0 = 1 for all i and ê n00 0 = 1. Write l i = 1 + a i1 t a ir t r (we set t 0 = 1) and i l i = 1 + α e α t α 1 1 tα r r (1 α n ). It appears as: π aff : A = a 11 a a n1 a 1r a nr (e α(a)) 1 α n The algebra of the affine chart of Symm n P r is DSym r n (K) (diagonal invariants of the symmetric group): the polynomials in the entries of A, invariants under row permutations. The e α are the elementary polynomials. The algebra DSym r n also contains analogues of the classical power sums and monomial functions and conversion algorithms.

10 The isomorphism problem Pb 1. Is Pπ an isomorphism Symm n P r = Chow(n,0, P r )? Neeman (1989): if char K = 0 or > n then Pπ is an isomorphism. Dalbec (1999): Pπ : Symm 2 P 2 Chow(2, P 2 ) is an isomorphism regardless of char K. E.B. (2002) For char K = 2, Pπ is an isomorphism iff r = 1 or n = 1 or r = 2 with n 3. For char K > 2, Pπ is an isomorphism iff n > char K or r = 1.

11 The isomorphism problem Idea of the proof: This can be established locally. Pπ is an isomorphism iff π aff (the map between affine charts) is an embedding iff the e α generate DSym r n(k). Use a finite generating set of DSym r n (over the integers) like Fleischmann s monomial functions or the Bergeron Lamontagne basis. Use projections DSym r n DSym r n 1 and DSymr n DSym r 1 n. Finish with a small number of small brute force computations.

12 From now on K = C Pb. 2: Compute the ideal of Chow(n,0, P r ) It is ker π = ideal of the algebraic relations between the ê α HDSym r+1 n (C) It is easier to get ker πaff = ideal of the algebraic relations between the e α DSym r n(c) because: A nice algorithm to produce all relations in given multidegree is known (Junker + Dalbec). The (multigraded) Hilbert series of DSym r n is known (it is the generating function for vector partitions). (Gessel Garsia 1979, Bergeron Lamontagne 2005) All is in favour of Hilbert driven Gröbner basis computations.

13 Compute the ideal of Chow(n,0, P r ) Two types of monomial orderings are interesting: 1. total degree order. Useful: A Gröbner basis for ker πaff (relations between elementary polynomials) provides a Gröbner basis for ker π by mere homogenization of its elements. 2. easy order, enjoying the structure of DSym r n. DSym r n is a free module of rank (n!) r 1 over a subalgebra = r Sym n. This provides small Gröbner bases and smaller (non Gröbner) generating sets. # small set of generators / # gb for nice order / # gb for total degree Symm n P r n = 2 n = 3 n = 4 n = 5 n = 6 r = 2 1/1/1 5/5/35 15/23/ /102/? 70/518/? r = 3 6/6/12 43/53/ /743/? r = 4 20/20/ 196/292/? Dalbec (1999) conjectured that the ideal of Chow(3,0, P 3 ) is generated in degree 4: true.

14 Foulkes Howe conjecture Foulkes (open) plethysm conjecture: (1950) h d h n h n h d is Schur positive for d n. Consider π = d π d : d=0 T n syms d V d=0 S d T n symv GL(V ) characters for the pieces of degree d: h n h d and h d h n

15 Foulkes Howe conjecture Foulkes (open) plethysm conjecture: (1950) h d h n h n h d is Schur positive for all d n. Consider π = d π d : d=0 T n syms d V d=0 S d T n symv GL(V ) characters for the pieces of degree d: h n h d and h d h n Howe s (stronger) conjecture(s) (1987) FH (i) π d is injective for all d n. FH (ii) π d is surjective for all d n.

16 Foulkes Howe conjecture Foulkes (open) plethysm conjecture: (1950) h d h n h n h d is Schur positive for all d n. Consider π = d π d : d=0 T n syms d V d=0 S d T n symv GL(V ) characters for the pieces of degree d: h n h d and h d h n Howe s (stronger) conjecture(s) (1987): FH (i) πd is injective for all d n. ( πn is bijective) ( No form of degree n vanishes on Chow(n,0, P r )) ( The degree n piece of HDSym r+1 n is generated by the fundamental homogeneous invariants ê α.) ( Chow(n,0, P r ) has no equation of degree n)

17 Foulkes Howe conjecture M. Brion (1997): FH conjecture (ii) is true for d >> n (with explicit lower bound depending on n and r) E.B. : FH conjectures (i) and (ii) are true for n = 3. E.B. (2002), J. Jacob (2004): FH conjecture (i) is true for n = 4.

18 Foulkes Howe conjecture M. Brion (1997): FH conjecture (ii) is true for d >> n (with explicit lower bound depending on n and r) E.B. : FH conjectures (i) and (ii) are true for n = 3. E.B. (2002), J. Jacob (2004): FH conjecture (i) is true for n = 4. Müller+Neunhöffer (2005): FH conjectures (i) and (ii) are false for n = 5.

19 Foulkes Howe conjecture M. Brion (1997): FH conjecture (ii) is true for d >> n (with explicit lower bound depending on n and r) E.B. : FH conjectures (i) and (ii) are true for n = 3. E.B. (2002), J. Jacob (2004): FH conjecture (i) is true for n = 4. Müller+Neunhöffer (2005): FH conjectures (i) and (ii) is false for n = 5. Anyway...when is π d injective? surjective?

20 Showing that FH (i) holds for fixed n: Toy example n = 2 To show: that the fundamental homogeneous invariants generate the degree n piece of HDSym r+1 n (C). Ex: n = 2, A = [ ] a1 a 2 b 1 b 2 The monomial functions = orbit sums of monomials Σ(a α 1 1 aα 2 2 bβ 1 1 bβ 2 (under row permutations of the matrix) are a linear basis for DSym. The decomposition 2 ) Σ(a 2 1 b 1b 2 ) = e 11 e 20 is obtained by applying the polarization operator to the key identity: 1 2 (a 1 d a 2 + b 1 d b2 )

21 Σ(a 2 1 b2 2 ) = a2 1 b2 2 + a2 2 b2 1 = e2 11 2e 20e 02 The key identity is also invariant under Column permutations!

22 Doubly symmetric functions Checking FH (i) for fixed n can be reduced to computations in the a 11 a 12 a 1n subspace of C[A] S n S n a : A = 21. of the elements.. a n1 a nn homogeneous of degree n w.r.t. the variables of each row and homogeneous of degree n w.r.t. the variables of each column. Ex: n = 3, linear bases are indexed by: Even the enumeration of these objects is difficult!

23 Conference: Diagonally symmetric polynomials and applications. October 15-19, 2007 Castro-Urdiales, Spain. Invited speakers: François Bergeron, Riccardo Biagioli, Ira Gessel, Francesco Vaccarino.

24 Conference: Diagonally symmetric polynomials and applications. October 15-19, 2007 Castro-Urdiales, Spain. Invited speakers: François Bergeron, Riccardo Biagioli, Ira Gessel, Francesco Vaccarino.