2-species exclusion processes and combinatorial algebras Sylvie - PowerPoint PPT Presentation

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives 2-species exclusion processes and combinatorial algebras Sylvie Corteel Arthur Nunge IRIF, LIGM March 2017

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Non commutative symmetric functions The algebra of noncommutative symmetric functions Sym is an algebra generalizing the symmetric functions. Its component of degree n has dimention 2 n − 1 . One can index its bases by compositions.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Non commutative symmetric functions The algebra of noncommutative symmetric functions Sym is an algebra generalizing the symmetric functions. Its component of degree n has dimention 2 n − 1 . One can index its bases by compositions. A composition of size n is a sequence of integers I = ( i 1 , i 2 , . . . , i r ) of sum n .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Non commutative symmetric functions The algebra of noncommutative symmetric functions Sym is an algebra generalizing the symmetric functions. Its component of degree n has dimention 2 n − 1 . One can index its bases by compositions. A composition of size n is a sequence of integers I = ( i 1 , i 2 , . . . , i r ) of sum n . Complete basis (analog of h λ ) For all n , define � S n = a j 1 a j 2 · · · a j n . 1 ≤ j 1 ≤ j 2 ≤···≤ j n

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Non commutative symmetric functions The algebra of noncommutative symmetric functions Sym is an algebra generalizing the symmetric functions. Its component of degree n has dimention 2 n − 1 . One can index its bases by compositions. A composition of size n is a sequence of integers I = ( i 1 , i 2 , . . . , i r ) of sum n . Complete basis (analog of h λ ) For all n , define � S n = a j 1 a j 2 · · · a j n . 1 ≤ j 1 ≤ j 2 ≤···≤ j n For any composition I = ( i 1 , i 2 , . . . , i r ), S I = S i 1 S i 2 · · · S i r .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Non commutative symmetric functions The algebra of noncommutative symmetric functions Sym is an algebra generalizing the symmetric functions. Its component of degree n has dimention 2 n − 1 . One can index its bases by compositions. A composition of size n is a sequence of integers I = ( i 1 , i 2 , . . . , i r ) of sum n . Complete basis (analog of h λ ) For all n , define � S n = a j 1 a j 2 · · · a j n . 1 ≤ j 1 ≤ j 2 ≤···≤ j n For any composition I = ( i 1 , i 2 , . . . , i r ), S I = S i 1 S i 2 · · · S i r . For example, S 2 ( a 1 , a 2 , a 3 ) = a 2 1 + a 1 a 2 + a 1 a 3 + a 2 2 + a 2 a 3 + a 2 3 .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Ribbon basis � ( − 1) l ( J ) − l ( I ) S J . R I = J � I For example, R 221 = S 221 − S 41 − S 23 + S 5 .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Ribbon basis � ( − 1) l ( J ) − l ( I ) S J . R I = J � I For example, R 221 = S 221 − S 41 − S 23 + S 5 . Polynomial realization � R I = w . Des( w )= I For example, R 221 ( a 1 , a 2 ) = a 1 a 2 a 1 a 2 a 1 + a 2 a 2 a 1 a 2 a 1 .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Tevlin’s bases In 2007 L. Tevlin defined the monomial ( M I ) and fundamental ( L I ) that are analog of the monomial basis and elementary basis of Sym . They both have binomial structure coefficients.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Tevlin’s bases In 2007 L. Tevlin defined the monomial ( M I ) and fundamental ( L I ) that are analog of the monomial basis and elementary basis of Sym . They both have binomial structure coefficients. Transition matrices The transition matrices between the ribbon basis and the fundamental basis of size 3 and 4 are:   1 . . . . 2 1 .   M 3 =   . . 1 .   1 . . .  1  . . . . . . . 3 2 1 1 . . . .     2 1 . . . . . .     . . 1 3 . 2 1 .   M 4 =   . . . . 1 . . .     . . . . . 2 1 .     . . . . . . 1 .   . . . . . . . 1

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 }

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 1.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 13.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 132.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 1322.

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 132. • GC( σ ) is the composition associated with the values of descents ( i.e. , the values k = σ i such that σ i > σ i +1 ) minus one. For σ = 25783641, GC( σ ) = .

Introduction Combinatorics of the 2-ASEP Generalization of Sym Conclusion and perspectives Statistics on permutations • Rec ( σ ) is the composition associated with the values of recoils ( i.e. , the values k such that k + 1 is on the left). For σ = 25783641, the recoils are { 1 , 4 , 6 } so Rec (25783641) = 132. • GC( σ ) is the composition associated with the values of descents ( i.e. , the values k = σ i such that σ i > σ i +1 ) minus one. For σ = 25783641, GC( σ ) = .