Algebraic combinatorics, representation theory, and Sage

Transcription

1 Algebraic combinatorics, representation theory, and Sage Anne Schilling, UC Davis ICERM, Providence January 8, 03 α α α 0 /6

2 Topics Representation theory: crystal bases, Lie theory, Hecke algebras, root systems, Coxeter groups,... Algebraic combinatorics: symmetric functions, (nonsymmetric) Macdonald polynomials, Demazure characters, tableaux,... Schubert calculus: Schubert polynomials, Kazhdan Lusztig polynomials, Demazure operators,... /6

3 Topics Representation theory: crystal bases, Lie theory, Hecke algebras, root systems, Coxeter groups,... Algebraic combinatorics: symmetric functions, (nonsymmetric) Macdonald polynomials, Demazure characters, tableaux,... Schubert calculus: Schubert polynomials, Kazhdan Lusztig polynomials, Demazure operators,... /6

4 Topics Representation theory: crystal bases, Lie theory, Hecke algebras, root systems, Coxeter groups,... Algebraic combinatorics: symmetric functions, (nonsymmetric) Macdonald polynomials, Demazure characters, tableaux,... Schubert calculus: Schubert polynomials, Kazhdan Lusztig polynomials, Demazure operators,... /6

5 Crystals B(Λ ) B(Λ + Λ ) /6

6 Axiomatic Crystals A U q (g)-crystal is a nonempty set B with maps wt B P e i, f i B B { } for all i I satisfying f i (b) = b e i (b ) = b wt(f i (b)) = wt(b) α i h i, wt(b) = ϕ i (b) ε i (b) if b, b B if f i (b) B Write b i b for b = f i (b) 4/6

7 Tensor products of crystals B(Λ ) B(Λ + Λ ) /6

8 Definition B, B crystals B B is B B as sets with Tensor products wt(b b ) = wt(b) + wt(b ) f i (b b f i (b) b if ε i (b) ϕ i (b ) ) = b f i (b ) otherwise b b ϕ i (b) ε i (b ) 6/6

9 Definition B, B crystals B B is B B as sets with Tensor products wt(b b ) = wt(b) + wt(b ) f i (b b f i (b) b if ε i (b) ϕ i (b ) ) = b f i (b ) otherwise b b ϕ i (b) ε i (b) ϕ i (b ) ε i (b ) 6/6

10 Definition B, B crystals B B is B B as sets with Tensor products wt(b b ) = wt(b) + wt(b ) f i (b b f i (b) b if ε i (b) ϕ i (b ) ) = b f i (b ) otherwise b b ϕ i (b) ε i (b) ϕ i (b ) ε i (b ) 6/6

11 Definition B, B crystals B B is B B as sets with Tensor products wt(b b ) = wt(b) + wt(b ) f i (b b f i (b) b if ε i (b) ϕ i (b ) ) = b f i (b ) otherwise b b +++ ϕ i (b) ε i (b) ϕ i (b ) ε i (b ) 6/6

12 Littlewood-Richardson rule in terms of crystals c ν λµ = LR coefficient V (λ) V (µ) cλµ ν V (ν) ν Theorem (Kashiwara-Nakashima) cλµ ν is the number of highest weight vectors in B(λ) B(µ) of weight ν. Crystals are now on lmfdb.org. Try to explore them there! Thematic Tutorial 7/6

13 Littlewood-Richardson rule in terms of crystals c ν λµ = LR coefficient V (λ) V (µ) cλµ ν V (ν) ν Theorem (Kashiwara-Nakashima) cλµ ν is the number of highest weight vectors in B(λ) B(µ) of weight ν. Crystals are now on lmfdb.org. Try to explore them there! Thematic Tutorial 7/6

14 Littlewood-Richardson rule in terms of crystals c ν λµ = LR coefficient V (λ) V (µ) cλµ ν V (ν) ν Theorem (Kashiwara-Nakashima) cλµ ν is the number of highest weight vectors in B(λ) B(µ) of weight ν. Crystals are now on lmfdb.org. Try to explore them there! Thematic Tutorial 7/6

15 Littlewood-Richardson rule in terms of crystals c ν λµ = LR coefficient V (λ) V (µ) cλµ ν V (ν) ν Theorem (Kashiwara-Nakashima) cλµ ν is the number of highest weight vectors in B(λ) B(µ) of weight ν. Crystals are now on lmfdb.org. Try to explore them there! Thematic Tutorial 7/6

16 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

17 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

18 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

19 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

20 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

21 Macdonald polynomials s λ [X ] Schur functions (basis for ring of symmetric functions) p λ [X ] power sum symmetric functions Macdonald symmetric functions H λ [X ; q, t] unique basis for ring of symmetric functions over Q((q, t)) s.t.: H λ [X ; q, t] = µ λ r λµ (q, t)s µ [X /( q)]; H λ [X ; q, t], H µ [X ; q, t] qt = 0 if λ µ where scalar product, qt is defined as p λ, p µ qt = z λ δ λµ ( q λ i )( t λ i ); i 3 H λ [X ; q, t], s (n) [X ] = t n(λ). Disadvantage: This is a very implicit definition! 8/6

22 Macdonald polynomials and crystals From the Ram-Yip formula in terms of crystals Theorem (Lenart) In types A and C, we have H µ (x; q, 0) = Using quantum alcove paths b B µ, B µ,... q charge(b) x weight(b) Theorem (Lenart, Naito, Sagaki, S, Shimozono) For general untwisted types H µ (x; q, 0) = q level(π) x weight(π) Π A(µ) 9/6

23 Macdonald polynomials and crystals From the Ram-Yip formula in terms of crystals Theorem (Lenart) In types A and C, we have H µ (x; q, 0) = Using quantum alcove paths b B µ, B µ,... q charge(b) x weight(b) Theorem (Lenart, Naito, Sagaki, S, Shimozono) For general untwisted types H µ (x; q, 0) = q level(π) x weight(π) Π A(µ) 9/6

24 Demazure crystals Littelmann conjectured/ Kashiwara proved that there is a subset B w (Λ) (Demazure crystal) of B(Λ) s.t. where b = D i D il u Λ b B w (Λ) i... i l is a reduced word of w D i b = 0 k hi,wt(b) fi kb if h i, wt(b) 0 k< hi,wt(b) ei kb if h i, wt(b) < 0. Corollary χ(b w (Λ)) is the Demazure character. 0/6

25 Demazure crystals Littelmann conjectured/ Kashiwara proved that there is a subset B w (Λ) (Demazure crystal) of B(Λ) s.t. where b = D i D il u Λ b B w (Λ) i... i l is a reduced word of w D i b = 0 k hi,wt(b) fi kb if h i, wt(b) 0 k< hi,wt(b) ei kb if h i, wt(b) < 0. Corollary χ(b w (Λ)) is the Demazure character. 0/6

26 Demazure crystals B s s (ω + ω ) Demazure crystal vertices given by {f af b(u ω +ω ) a, b 0} /6

27 Kyoto path model, affine Demazure and KR crystals Theorem (Kyoto path model) Highest weight infinite-dimensional crystal in terms of semi-infinite tensor product B(sΛ 0 ) B r,s B r,s B(sΛ i ) Theorem (Fourier, S, Shimozono; S, Tingley) Affine Demazure crystal D(ω r, s) B r,s u sλ0 /6

28 Kyoto path model, affine Demazure and KR crystals Theorem (Kyoto path model) Highest weight infinite-dimensional crystal in terms of semi-infinite tensor product B(sΛ 0 ) B r,s B r,s B(sΛ i ) Theorem (Fourier, S, Shimozono; S, Tingley) Affine Demazure crystal D(ω r, s) B r,s u sλ0 /6

29 Affine Demazure crystals Demazure 0-arrows non-demazure 0-arrows (not all given) 3/6

30 Macdonald polynomials and Demazure characters Theorem (Ion) For simply-laced or twisted root system the (nonsymmetric) Macdonald polynomials at t = 0 equal Demazure characters: H λ [X ; q, 0] = χ(b wλ (Λ 0 )) To do! Generalize to untwisted types. 4/6

31 Macdonald polynomials and Demazure characters Theorem (Ion) For simply-laced or twisted root system the (nonsymmetric) Macdonald polynomials at t = 0 equal Demazure characters: H λ [X ; q, 0] = χ(b wλ (Λ 0 )) To do! Generalize to untwisted types. 4/6

32 Sage Sage s mission: Creating a viable free open source alternative to Magma TM, Maple TM, Mathematica TM, and Matlab TM. free open source anyone can contribute! Design: Sage is build around Python (general-purpose programming language) 5/6

33 Sage Sage s mission: Creating a viable free open source alternative to Magma TM, Maple TM, Mathematica TM, and Matlab TM. free open source anyone can contribute! Design: Sage is build around Python (general-purpose programming language) 5/6

34 Sage Sage s mission: Creating a viable free open source alternative to Magma TM, Maple TM, Mathematica TM, and Matlab TM. free open source anyone can contribute! Design: Sage is build around Python (general-purpose programming language) 5/6

35 Sage Sage s mission: Creating a viable free open source alternative to Magma TM, Maple TM, Mathematica TM, and Matlab TM. free open source anyone can contribute! Design: Sage is build around Python (general-purpose programming language) 5/6

36 Sage Sage s mission: Creating a viable free open source alternative to Magma TM, Maple TM, Mathematica TM, and Matlab TM. free open source anyone can contribute! Design: Sage is build around Python (general-purpose programming language) 5/6

37 History: Sage Sage project began by William Stein in 005 SAGE= Software for Arithmetic Geometry Experimentation Quickly expanded beyond number theory; attracted more users, developers, funding sagenb.org now has over 90,000 accounts Sage-combinat: To improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area. Try it! Use Sage online Install Sage 6/6

38 History: Sage Sage project began by William Stein in 005 SAGE= Software for Arithmetic Geometry Experimentation Quickly expanded beyond number theory; attracted more users, developers, funding sagenb.org now has over 90,000 accounts Sage-combinat: To improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area. Try it! Use Sage online Install Sage 6/6

39 History: Sage Sage project began by William Stein in 005 SAGE= Software for Arithmetic Geometry Experimentation Quickly expanded beyond number theory; attracted more users, developers, funding sagenb.org now has over 90,000 accounts Sage-combinat: To improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area. Try it! Use Sage online Install Sage 6/6

40 History: Sage Sage project began by William Stein in 005 SAGE= Software for Arithmetic Geometry Experimentation Quickly expanded beyond number theory; attracted more users, developers, funding sagenb.org now has over 90,000 accounts Sage-combinat: To improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area. Try it! Use Sage online Install Sage 6/6

41 History: Sage Sage project began by William Stein in 005 SAGE= Software for Arithmetic Geometry Experimentation Quickly expanded beyond number theory; attracted more users, developers, funding sagenb.org now has over 90,000 accounts Sage-combinat: To improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area. Try it! Use Sage online Install Sage 6/6