On the growth of the Kronecker coefficients.

Transcription

1 On the growth of the Kronecker coefficients. May 15, 2017 Algebraic Combinatorixx, Banff 2017

2 Summary

3 What are the Kronecker coefficients? The representation theory of the general lineal group. The representation theory of the symmetric group. Symmetric functions.

4 The representation theory of the general lineal group A representation of GL(V ) is a vector space W together with a group homomorphism ρ : GL(V ) GL(W ). The character of ρ is a function from GL(V ) C that to any matrix A it associates the trace of ρ(a). We assume that our representions are polynomial.

5 Defining representation Let {e 1, e 2,, e d } a eigenbasis for A, that we can assume to be diagonalizable. Ae k = α k e k The identity φ(a) = A defines a representation GL(V ) GL(V ) of degree d. Its character is just the trace of A, i.e., the sum of the eigenvalues of A: ch(v ) = s 1 (α 1, α 2,, α d ) = α 1 + α α d

6 Symmetric Powers, Sym k V There is a representation of GL(V ) on Sym k (V ) induced by the diagonal action of GL(V ) on Sym k V. Sym k : GL(V ) GL(Sym k V )) A basis for Sym k V is given by the symmetric products with i 1 i 2, i k. e i1 e i2 e ik

7 The character of Sym k V Then, A(e i1 e i2 e ik ) = A(e i1 ) A(e i2 ) A(e ik ) = (α i1 α i2 α ik )e i1 e i2 e ik The trace of this homomorphism is the sum of all different monomials α i1 α i2 α ik of degree k: ch(sym k V ) = s (k) (α 1, α 2,, α d )

8 The exterior powers, k V, with k dim V Let A act on k V diagonally. This defines a representation k V : GL(V ) GL( k (V )) A(e i1 e i2 e ik ) = A(e i1 ) A(e i2 ) A(e ik ) = (α i1 α i2 α ik )e i1 e i2 e ik with strictly increasing subindices. The character of the representation k V is k ch( V ) = s(1 k)(α 1, α 2,, α d )

9 Weyl s construction To any partition λ of length dim(v ) Weyl associated an irrep of GL(V ) inside of V using Young symmetrizers. A basis of W λ is labelled by all semistandard filling of λ. The degree of the representation W λ (V ) is the number of semistandard filling of λ with entries 1, 2,, dim(v ).

10 The restriction of representations The product GL(V ) GL(W ) can be embedded into GL(V W ) : Let A GL(V ) and B GL(W ), (A, B) acts on V W : (A, B)(v w) = Av Bw. That is, GL(V ) GL(W ) GL(V W ) W λ (V W ) The restriction of W λ (V W ) to GL(V ) GL(W ) defines a (usually reducible) representation of GL(V ) GL(W ).

11 The Kronecker coefficients The restriction of representations is usually reducible W λ GL(V W ) (V W ) GL(V ) GL(W ) = g µ,ν,λ W µ (V ) W ν (W ) The g λ,µ,ν are the Kronecker coefficients. Being multiplicities are non-negative integers.

12 The Character of V W Let {e 1, e 2, e d1 } and {f 1, f 2, f d2 } be eigenbases for A and B, with eigenvalues α i and β j. Then e i f j is an eigenbasis for V W with eigenvalues α i β j (A, B)(e i f j ) = A(e i ) B(f j ) = α i β j (e i f j ) Therefore, ch(v W ) = s 1 (α 1 β 1, α 1 β 2, α d1 β d2 )

13 ch(w λ (V W )) Since ch(w λ (V W )) is the Schur function s λ evaluated at the eigenvalues for the representation V W ch(w λ (V W )) = s λ (α 1 β 1, α 1 β 2, α d1 β d2 )

14 Summary The isormorphism of representations W λ GL(V W ) (V W ) GL(V ) GL(W ) = g µ,ν,λ W µ (V ) W ν (W ) translates into the identity of symmetric functions: s λ (α 1 β 1, α 1 β 2, α d1 β d2 ) = µ,ν g µ,ν,λ s µ (α 1,, α d1 )s ν (β 1,, β d2 )

15 The product of two alphabets If A = α α d1 and B = β β d2 then Therefore, AB = α 1 β 1 + α 1 β 2 + α d1 β d2 s λ (α 1 β 1, α 1 β 2, α d1 β d2 ) = µ,ν g µ,ν,λ s µ (α 1,, α d1 )s ν (β 1,, β d2 ) Becomes s λ [AB] = g µ,ν,λ s µ [A]s ν [B] µ,ν

16 The stretched Kronecker quasi polynomial Q λ,µ,ν (t) Let λ, µ and ν be three partitions of n. The stretched Kronecker quasi polynomial Q λ,µ,ν (t) is defined as Q λ,µ,ν (t) = g tλ,tµ,tν Thm [Meinrenken-Sjamaar 1999, Mulmuley 2007] The function Q λ,µ,ν (t) is indeed a quasi polynomial.

17 Some examples of Kronecker quasi-polynomials For λ = (1, 1), µ = (1, 1), ν = (1, 1) 1, if t 0 mod 2 g λ,µ,ν = 0, Q λ,µ,ν (t) = 0, if t 1 mod 2 For λ = (2, 1), µ = (2, 1), ν = (2, 1) g λ,µ,ν = 2, Q λ,µ,ν (t) = (t + 2)/2, if t 0 mod 2 (t + 1)/2, if t 1 mod 2

18 Lineal growth A related result Thm [Briand, Rattam, R] If the three indexing partitions have long first parts, and we add t cells to each of their first two rows, then, the Kronecker coefficients are described by a linear quasipolynomial of period 2 on t.

19 The remaining rows Thm [Colmenarejo-R:] The reduced Kronecker coefficient indexed by (k), (k a ), (k a ) count the number of plane partitions of k fitting inside a 2 a rectangle. The corresponding quasi-polynomial has degree 2a 1 and period lcm(1, 2,, a + 1).

20 A cubic quasi polynomial Thm [Colmenarejo-R:] F 3 6 g k k 2, k 2 g k k 2, k k 3 1 6k 2 2 3k 1 k 0 mod k 3 1 6k k 5 18 k 1 mod k 3 1 6k 2 2 3k 8 9 k 2 mod k 3 1 6k k 1 2 k 3 mod k 3 1 6k 2 2 3k 7 9 k 4 mod k 3 1 6k k 7 18 k 5 mod 6

21 Why is this interesting? Asymptotic of the Kronecker coefficients. This quasi polynomial gives us information about the kind of objects that can be counted by the Kronecker coefficients.

22 An interesting example of Kronecker quasi polynomial Let λ = (6, 6), µ = (7, 5), and ν = (6, 4, 2) be partitions. Briand, Orellana, R showed that Mulmuley s saturation hypothesis does not hold. For even values of t whereas for odd values of t Q λ,µ,ν = 1 (t + 2). 2 Q λ,µ,ν = 1 (t + 2). 2

23 Ron King s observation In 2009, in Sevilla, Ron King observed that for λ = (6, 6), µ = (7, 5), and ν = (6, 4, 2) Q λ,µ,ν (m) ( 1) m Q λ,µ,ν ( m) Ehrhart reciprocity fails. Q λ,µ,ν can NOT count the number of integral points on any dilated rational polytope.

24 The work of Baldoni and Vergnes On their paper Computations of dilated Kronecker coefficients Baldoni and Vergnes found an algorithm that computes the dilated Kronecker coefficients, and implemented it on Maple. For partitions of bounded length it runs in polynomial time. There are also plenty of interesting examples.

25 Only Don Quijote would be find it realistic to try to describe combinatorial formulas for the Kronecker quasi-polynomials.

26 Garsia, Wallach, Xin, Zabrocki: s (d,d),(d,d),λ = ((all parts odd or all parts even)) Brown, van Willigenburg, Zabrocki: g (d,d),(d+k,d k),λ = 1{ if k λ 2 (mod 2) and λ 2 k and zero otherwise

27 Ron King s conjectures In his Sevilla talk, Ron King proposed a series of conjectures concerning the Kronecher quasi polynomial Q (n 1,1),µ,µ Q ν λµ (t) with λ =(n 1, 1) and µ = ν Cases with a>0, a>b>0 and a>b>c>0 as appropriate for t =0, 1,...,12 µ = ν a aa ab aaa aab, abb abc Conjecture The results are independent of a, b, c (taken from his slides) Seville-2009 p.38

28 Ron King s conjectures Q ν λµ (t) with λ =(n 1, 1) and µ = ν Cases with a > b > c > d > e > 0 for t =0, 1,...,10 µ = ν aaaaa abbbb aabbb abccc abbcc abcdd abcde Conjecture The results are independent of a, b, c, d, e The cases abbbb and aaaab are identical, etc. (taken from his slides) Seville-2009 p

29 Ron King s conjectures Q ν λµ (t) with λ =(n 1, 1) and µ = ν Cases with a > b > c > d > 0 for t =0, 1,...,10 µ = ν aaaa abbb aabb abcc abcd Conjecture The results are independent of a, b, c, d The cases abbb and aaab are identical The cases abcc, abbc and aabc are identical (taken from his slides)

30 A combinatorial proof for Ron King s conjectures The starting point of our work will be a combinatorial interpretation for Kronecher quasi polynomial Q (n 1,1),µ,µ. We will explore where does it takes us.