Cyclic inclusion/exclusion

Transcription

1 Cyclic inclusion/exclusion Valentin Féray LaBRI, CNRS, Bordeaux Journées Combinatoire Algébrique du GDR-IM, Rouen Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

2 Introduction Context What is this talk about? To a bipartite graph, we will associate a formal series: i,j p i p j q max(i,j) Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

3 Introduction Context What is this talk about? To a bipartite graph, we will associate a formal series: i,j p i p j q max(i,j) Question: which linear combination of graphs are sent to 0? Example : + 0. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

4 Introduction Context What is this talk about? To a bipartite graph, we will associate a formal series: i,j p i p j q max(i,j) Question: which linear combination of graphs are sent to 0? Example : + 0. Motivation : computation of irreducible character values of symmetric groups. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

5 Introduction Context Definition of N Let p, q be two infinite set of variables. G = N(G) = General formula: N(G) = Q[p,q], Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

6 Introduction Context Definition of N Let p, q be two infinite set of variables. G = N(G) = General formula: N(G) = Q[p,q], Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

7 Introduction Context Definition of N Let p, q be two infinite set of variables. G = i j N(G) = i,j p i p j General formula: N(G) = p ϕ( ) ϕ:v N V Q[p,q], Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

8 Introduction Context Definition of N Let p, q be two infinite set of variables. max(i, j) j G = i N(G) = i,j j p i p j q j q max(i,j) General formula: N(G) = p ϕ( ) q ψ( ) ϕ:v N V V Q[p,q], with ψ( ) = max ϕ( ). Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

9 Kernel of our algebra morphism N as an algebra morphism Let BG be the Q vector space of linear combination of bipartite graphs. It is an algebra G G = G G. N defines a morphism of algebra BG Q[p,q] G p ϕ( ) ϕ:v N V V q ψ( ), Question What is the kernel of N? Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

10 Kernel of our algebra morphism Cyclic inclusion/exclusion Consider a bipartite graph G endowed with an oriented cycle C. We define the following element of BG: A G,C = ( 1) E G \ E Example : E E (C) + Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

11 Kernel of our algebra morphism Kernel of N Proposition N(A G,C ) = 0. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

12 Kernel of our algebra morphism Kernel of N Proposition N(A G,C ) = 0. Sketch of proof. We look at the contribution of a fixed function ϕ : V (G) N Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

13 Kernel of our algebra morphism Kernel of N Proposition N(A G,C ) = 0. Sketch of proof. We look at the contribution of a fixed function ϕ : V (G) N Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

14 Kernel of our algebra morphism Kernel of N Proposition N(A G,C ) = 0. Sketch of proof. We look at the contribution of a fixed function ϕ : V (G) N Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

15 Kernel of our algebra morphism Presentation of the next few slides We will show that N defines an injective morphism BG/ A G,C Z[[p,q]]. Method: Construct a family of graphs G I. Show that G I is a generating family in the quotient BG/ A G,C. Prove that N(G I ) is linearly independant in Z[[p,q]]. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

16 Kernel of our algebra morphism A generating family Definition Let I = (i 1,i,...,i r ) be a composition. Define G I as the following bipartite graph: i1-1 black vertices i -1 black vertices i_r -1 black vertices Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

17 Kernel of our algebra morphism A generating family Definition Let I = (i 1,i,...,i r ) be a composition. Define G I as the following bipartite graph: i1-1 black vertices i -1 black vertices i_r -1 black vertices Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

18 Kernel of our algebra morphism A generating family Definition Let I = (i 1,i,...,i r ) be a composition. Define G I as the following bipartite graph: i1-1 black vertices i -1 black vertices i_r -1 black vertices Proposition {G I,I composition} is a linear generating set of BG/I. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

19 Kernel of our algebra morphism Idea of proof Proposition {G I,I composition} is a linear generating set of BG/I. graph G I Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

20 Kernel of our algebra morphism Idea of proof Proposition {G I,I composition} is a linear generating set of BG/I. Consider a graph G G I Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

21 Kernel of our algebra morphism Idea of proof Proposition {G I,I composition} is a linear generating set of BG/I. There is a graph G 0 with an oriented cycle C such that G 0 \E (C) = G Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

22 Kernel of our algebra morphism Idea of proof Proposition {G I,I composition} is a linear generating set of BG/I. Consider a graph G G I. Lemma: There is a graph G 0 with an oriented cycle C such that: G 0 \E (C) = G Consequence : in BG/I, G = linear combination of bigger graphs. we iterate until we obtain a linear combination of G I s. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

23 Kernel of our algebra morphism Independence Lemma The N(G I ), where I runs over all compositions are linearly independent. Gradation: N(G I ) is a homogenous polynomial of degree r = l(i) in p and of total degree n = I. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

24 Kernel of our algebra morphism Independence Lemma The N(G I ), where I runs over compositions of length r and size n are linearly independent. Consider M GI (p 1,p,...,p r,q 1,q,...,q r ) (we truncate the alphabets to r = l(i) = V (G I ) variables) Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

25 Kernel of our algebra morphism Independence Lemma The N(G I ), where I runs over compositions of length r and size n are linearly independent. Consider M GI (p 1,p,...,p r,q 1,q,...,q r ) (we truncate the alphabets to r = l(i) = V (G I ) variables) We will consider only p-square free monomials. As total degree in p is r, they are: T J = p 1 q j p q j 1 p r q jr 1 r, where J is a composition of n and length r. In N(G I ), they correspond to bijections ϕ : V (G I ) {1,...,r}. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

26 Kernel of our algebra morphism Independence Lemma The N(G I ), where I runs over compositions of length r and size n are linearly independent. 1 r 1 1 r 1 r M GI = T I Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

27 Kernel of our algebra morphism Independence Lemma The N(G I ), where I runs over compositions of length r and size n are linearly independent M GI = T I + J = I =n, l(j)=l(i)=r J I c I,J T J + non-p-square-free terms stands for the right-dominance order. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

28 Other contexts One set of variables We considered a morphism BG Q[[p,q]] Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

29 Other contexts One set of variables It factorizes via BG BG/ A G,C Q[[p,q]] Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

30 Other contexts One set of variables Same kernel if we identify p i and q i! BG BG/ A G,C Q[[p,q]] p i x i q i x i Q[[x]] Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

31 Other contexts One set of variables We can describe the image BG BG/ A G,C Q[[p,q]] QSym(x) Q[[x]] p i x i q i x i QSym(x): ring of quasi-symmetric function in x. Example: M 1, (x 1,x,x 3 ) = x 1 x + x 1x 3 + x x 3. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

32 Other contexts One set of variables We can describe the image BG BG/ A G,C Q[[p,q]] QSym(x) Q[[x]] p i x i q i x i QSym(x): ring of quasi-symmetric function in x. Example: M 1, (x 1,x,x 3 ) = x 1 x + x 1x 3 + x x 3. BG QSym is a Hopf algebra morphism! Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

33 Other contexts Coproduct on BG = Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

34 Other contexts Some variants N (G) := ϕ:v G N (, ) E G ϕ( ) ϕ( ) v V G x ϕg. Then Ker(N ) = Ker(N). Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

35 Other contexts Some variants N (G) := ϕ:v G N (, ) E G ϕ( ) ϕ( ) v V G x ϕg. Then Ker(N ) = Ker(N). The definition above is naturally extended to acyclic directed graphs. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

36 Other contexts Some variants N (G) := ϕ:v G N (, ) E G ϕ( ) ϕ( ) v V G x ϕg. Then Ker(N ) = Ker(N). The definition above is naturally extended to acyclic directed graphs. One can also consider labeled graphs and polynomials in non-commutative variables (QSym is replaced by WQSym). N(G) = a f(1)... a f(n), f:[n] G f ր where the a s are non-commutative variables. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

37 Other contexts Does cyclic inclusion/exclusion always span the kernel? Same method as before: 1 Construct a family of (labelled/unlabelled) (bipartite/directed acyclic) graphs. Show that it is a generating family in the quotient G / A G,C. 3 Prove that the corresponding functions are linearly independant. 3 is hard in non-commutative setting. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

38 Other contexts The family of graphs in the bipartite labelled setting We consider set compositions (or ordered set-partitions) I. Example: I = We associate the following graph G I : Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

39 Other contexts The family of graphs in the bipartite labelled setting We consider set compositions (or ordered set-partitions) I. Example: I = We associate the following graph G I : Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

40 Other contexts The family of graphs in the bipartite labelled setting We consider set compositions (or ordered set-partitions) I. Example: I = We associate the following graph G I : Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

41 Other contexts I need some help here! Proposition The graphs G I generate the quotient G / A G,C Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

42 Other contexts I need some help here! Proposition The graphs G I generate the quotient G / A G,C Conjecture N(G I ), where I runs over all set compositions, is a basis of WQSym. Equivalently, the A G,C span the kernel of N. Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16

43 Other contexts I need some help here! Proposition The graphs G I generate the quotient G / A G,C Conjecture N(G I ), where I runs over all set compositions, is a basis of WQSym. Equivalently, the A G,C span the kernel of N. Thanks for listening! Valentin Féray (LaBRI, CNRS) Cyclic inclusion/exclusion Rouen, / 16