A zoo of Hopf algebras

Transcription

1 Non-Commutative Symmetric Functions I: A zoo of Hopf algebras Mike Zabrocki York University Joint work with Nantel Bergeron, Anouk Bergeron-Brlek, Emmanurel Briand, Christophe Hohlweg, Christophe Reutenauer, Mercedes Rosas and others...

2 Partitions Unordered lists of non-negative integers

3 Partitions Unordered lists of non-negative integers

4 Partitions Unordered lists of non-negative integers

5 Partitions Unordered lists of non-negative integers

6 Create an algebra

7 Create an algebra linearly spanned by partitions

8 Create an algebra linearly spanned by partitions Commutative product

9 Create an algebra linearly spanned by partitions Commutative product Graded by size of partitions

10 Create an algebra linearly spanned by partitions Commutative product Graded by size of partitions

11 Create an algebra linearly spanned by partitions Commutative product Graded by size of partitions linear span of partitions of size

12 Create an algebra linearly spanned by partitions Commutative product Graded by size of partitions linear span of partitions of size

13 Define a commutative product

14 Define a commutative product

15 Define a commutative product

16 Define a commutative product

17 properties of this algebra Commutative and graded

18 properties of this algebra Commutative and graded

19 properties of this algebra Commutative and graded

20 Properties of this algebra Freely generated by building blocks of rows

21 Properties of this algebra Freely generated by building blocks of rows

22 Properties of this algebra Freely generated by building blocks of rows

23 properties of this algebra

24 properties of this algebra Bilinear

25 properties of this algebra Bilinear

26 properties of this algebra Bilinear Isomorphic to the free polynomial algebra with one generator at each degree

27 properties of this algebra Bilinear Isomorphic to the free polynomial algebra with one generator at each degree

28 properties of this algebra Bilinear Isomorphic to the free polynomial algebra with one generator at each degree

29 From combinatorial object to algebra partitions

30 From combinatorial object to algebra some commutative product partitions

31 From combinatorial object to algebra some commutative product Algebra partitions

32 From combinatorial object to algebra another commutative product Algebra partitions

33 A different commutative product And yet the algebra which arises is isomorphic to

34 A different commutative product And yet the algebra which arises is isomorphic to

35 This algebra is special

36 This algebra is special Just about any algebra with basis indexed by partitions is Isomorphic: e.g. symmetric functions, representation ring of symmetric group, ring of characters of GLn modules, cohomology rings of the grassmannians, etc.

37 This algebra is special Just about any algebra with basis indexed by partitions is Isomorphic: e.g. symmetric functions, representation ring of symmetric group, ring of characters of GLn modules, cohomology rings of the grassmannians, etc. Has a Hopf algebra structure product + coproduct + antipode which all interact nicely with each other

38 What is a Hopf algebra?

39 What is a Hopf algebra? Start with a Bialgebra

40 What is a Hopf algebra? Start with a Bialgebra Product Coproduct

41 What is a Hopf algebra? Start with a Bialgebra Product Coproduct with unit and counit

42 What is a Hopf algebra? Start with a Bialgebra Product Coproduct with unit and counit and an antipode map

43 What is a Hopf algebra? Start with a Bialgebra this diagram commutes

44 What is so good about a Hopf algebra? The graded kind associated with combinatorial objects have lots of structure There seems to be just one graded combinatorial hopf algebra for each type of combinatorial object Many of the combinatorial operations are reflected in the algebraic structure

45 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions

46 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions Non-commutative symmetric functions will be non-commutative and generated by one element at each degree

47 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions Non-commutative symmetric functions will be non-commutative and generated by one element at each degree compositions

48 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions Non-commutative symmetric functions will be non-commutative and generated by one element at each degree compositions concatenation product

49 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions Non-commutative symmetric functions will be non-commutative and generated by one element at each degree compositions concatenation product

50 CHA s in the mid-90s the symmetric functions are commutative and generated by one element at each degree partitions Non-commutative symmetric functions will be non-commutative and generated by one element at each degree compositions concatenation product I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon

51 CHA s in the mid-90s partitions

52 CHA s in the mid-90s partitions compositions

53 CHA s in the mid-90s partitions compositions

54 CHA s in the mid-90s partitions compositions GKLLRT ( 95)

55 CHA s in the mid-90s partitions compositions GKLLRT ( 95) compositions

56 CHA s in the mid-90s partitions compositions GKLLRT ( 95) compositions Commutative algebra of Quasi-symmetric functions

57 CHA s in the mid-90s partitions compositions GKLLRT ( 95) compositions Commutative algebra of Quasi-symmetric functions Gessel ( 84)

58 CHA s in the mid-90s

59 CHA s in the mid-90s

60 CHA s in the mid-90s Malvenuto-Reutenauer ( 95) permutations

61 CHA s in the 90s+ uniform block permutations Aguiar-Orellana 05

62 CHA s in the 90s+ Tableaux uniform block permutations Poirier-Reutenauer 95 Aguiar-Orellana 05

63 CHA s in the 90s+ binary trees Connes-Kreimer 98 Grossman-Larson 89 Loday-Ronco 98 Tableaux uniform block permutations Poirier-Reutenauer 95 Aguiar-Orellana 05

64 A zoo of Hopf Algberas Hopf s graded

65 A goal of research on graded Hopf algebras? One Goal of determining the Hopf algebras associated to combinatorial objects is to try and arrive at a classification theorem for Graded combinatorial Hopf algebras A CLassification Theorem

66 A goal of research on graded Hopf algebras? One Goal of determining the Hopf algebras associated to combinatorial objects is to try and arrive at a classification theorem for Graded combinatorial Hopf algebras A CLassification Theorem 7 Marcelo Aguiar Species CHAs

67 Hopf algebras of set partitions/compositions

68 Hopf algebras of set partitions/compositions

69 Hopf algebras of set partitions/compositions Set composition definition:

70 Hopf algebras of set partitions/compositions Set composition definition: Set partition definition:

71 Another type of non-commutative symmetric functions Invariants under the left action on the polynomial ring Free algebra generated by one element at each degree Invariants under the left action on the non-comm poly ring Wolf 1936, Rosas-Sagan 03

72 Monomial set partition For each monomial Associate a set partition

73 Example:

74 Example:

75 Example:

76 Example: This is a non commutative polynomial for a given

77 Example: This is a non commutative polynomial for a given Consider the elements to be the object by letting the number of variables in

78 Combinatorial rule for product

79 Combinatorial rule for product

80 Combinatorial rule for product

81 Coproduct Inspiration:

82 Coproduct Inspiration:

83 Coproduct Inspiration:

84 Definition of Non-commutative Co-commutative Hopf algebra of set partitions

85 Analogy between Sym and NCSym

86 Properties of NCSym

87 Properties of NCSym Non-commutative and co-commutative

88 Properties of NCSym Non-commutative and co-commutative Has bases analogous to power, elementary, homogeneous, monomial symmetric functions in the algebra of Sym (Rosas-Sagan 03, Bergeron-Brlek)

89 Properties of NCSym Non-commutative and co-commutative Has bases analogous to power, elementary, homogeneous, monomial symmetric functions in the algebra of Sym (Rosas-Sagan 03, Bergeron-Brlek) The dimension of the part of degree n are the bell numbers 1, 1, 2, 5, 15, 52, 203, 877, 4140,...

90 Some questions to ask

91 Some questions to ask what is a fundamental basis of this space (analogue of Schur basis)?

92 Some questions to ask what is a fundamental basis of this space (analogue of Schur basis)? What is the connection with representation theory?

93 Some questions to ask what is a fundamental basis of this space (analogue of Schur basis)? What is the connection with representation theory? What is the structure of this algebra?

94 Some questions to ask what is a fundamental basis of this space (analogue of Schur basis)? What is the connection with representation theory? What is the structure of this algebra? What is the relationship with the other non-commutative symmetric functions? (NSym) compositions set partitions

95 The connection between NSym and NCSym NCSym has graded dimensions 1, 1, 2, 5, 15, 52, 203, 877, 4140,... NSym has graded dimensions 1, 1, 2, 4, 8, 16, 32, 64, 128,...

96 The connection between NSym and NCSym NCSym has graded dimensions 1, 1, 2, 5, 15, 52, 203, 877, 4140,... NSym has graded dimensions 1, 1, 2, 4, 8, 16, 32, 64, 128,... There exists a Hopf morphism

97 Families of morphisms

98 Families of morphisms

99 One last open question

100 One last open question What is the Hopf algebra of lions?

101 One last open question What is the Hopf algebra of lions?

102 One last open question What is the Hopf algebra of lions? Rowr!