Non-Commutative Symmetric Functions I A zoo of Hopf...
Transcript of Non-Commutative Symmetric Functions I A zoo of Hopf...
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A zoo of Hopf algebras
Mike ZabrockiYork University
Non-Commutative Symmetric Functions I:
I will present an overview of what Combinatorial Hopf Algebras (CHAs) are about by introducing definitions and examples. I will try to show how examples of CHAs are related to each other and where they can appear in the literature before they are recognized as CHAs. I will also give some examples where these objects appear in other areas of mathematics.
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Partitions
Unordered lists of non-negative integers
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Partitions
Unordered lists of non-negative integers
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Partitions
Unordered lists of non-negative integers
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Partitions
Unordered lists of non-negative integers
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Create an algebra
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Create an algebra
linearly spanned by partitions
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Create an algebra
linearly spanned by partitions
Commutative product
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Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
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Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
![Page 11: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/11.jpg)
Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
linear span of partitions of size
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Create an algebra
linearly spanned by partitions
Commutative product
Graded by size of partitions
linear span of partitions of size
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Define a commutative product
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Define a commutative product
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Define a commutative product
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Define a commutative product
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properties of this algebra
Commutative and graded
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properties of this algebra
Commutative and graded
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properties of this algebra
Commutative and graded
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Properties of this algebraFreely (commutative) generated by building blocks of rows
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Properties of this algebraFreely (commutative) generated by building blocks of rows
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Properties of this algebraFreely (commutative) generated by building blocks of rows
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properties of this algebra
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properties of this algebra
Bilinear
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properties of this algebra
Bilinear
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properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
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properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
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properties of this algebra
Bilinear
Isomorphic to the free polynomial algebra with one generator at each degree
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From combinatorial object to algebra
partitions
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From combinatorial object to algebra
partitions
some commutative
product
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From combinatorial object to algebra
partitions
some commutative
product
Algebra
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From combinatorial object to algebra
partitions
Algebraanother commutative
product
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A different commutative product
And yet the algebra which arises isisomorphic to
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A different commutative product
And yet the algebra which arises isisomorphic to
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This algebra is special
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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.
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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 structureproduct + coproduct + antipodewhich all interact nicely with each other
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What is a Hopf algebra?
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What is a Hopf algebra?Start with a Bialgebra
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What is a Hopf algebra?
Product
Coproduct
Start with a Bialgebra
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What is a Hopf algebra?
Product
Coproduct
with unit
and counit
Start with a Bialgebra
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What is a Hopf algebra?
Product
Coproduct
with unit
and an antipode map
and counit
Start with a Bialgebra
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What is a Hopf algebra?
this diagram commutes
Start with a Bialgebra
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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
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What is so good about a Hopf algebra?
There is “usually” an internal product structure and on some bases of some algebras this is hard (but important) to explain
Aguiar-Bergeron-Sottile saysA combinatorial Hopf algebra is a graded connected Hopf algebra with a multiplicative linear function.
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
partitions
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
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CHA’s in the mid-90sthe symmetric functions are
commutative and generated by oneelement at each degree
Non-commutative symmetric functionswill be non-commutative and generated
by one element at each degree
partitions
compositions concatenationproduct
I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon
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CHA’s in the mid-90s
partitions
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CHA’s in the mid-90s
compositions
partitions
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CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
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CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
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CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositions
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CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositions
Commutative algebraof Quasi-symmetric functions
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CHA’s in the mid-90s
compositions
partitions
GKLLRT (ʻ95)
compositionsGessel (ʻ84)
Commutative algebraof Quasi-symmetric functions
![Page 59: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/59.jpg)
CHA’s in the mid-90s
FQSym
![Page 60: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/60.jpg)
CHA’s in the mid-90s
FQSym
![Page 61: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/61.jpg)
CHA’s in the mid-90s
permutations
Malvenuto-Reutenauer (ʻ95)
FQSym
![Page 62: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/62.jpg)
CHA’s in the 90s+
![Page 63: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/63.jpg)
CHA’s in the 90s+
uniform block permutations
Aguiar-Orellana ‘05
![Page 64: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/64.jpg)
CHA’s in the 90s+
uniform block permutations
Aguiar-Orellana ‘05
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8
9
10
11
12
13
Poirier-Reutenauer ‘95
Tableaux
![Page 65: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/65.jpg)
CHA’s in the 90s+binary trees Connes-Kreimer ’98
Grossman-Larson ‘89
Loday-Ronco ‘98
uniform block permutations
Aguiar-Orellana ‘05
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6
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8
9
10
11
12
13
Poirier-Reutenauer ‘95
Tableaux
![Page 66: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/66.jpg)
CHA’s in the 90s+CHA’s in the 90s+
![Page 67: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/67.jpg)
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
![Page 68: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/68.jpg)
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
Packed Words
Hivert ‘99
Set Compositions
![Page 69: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/69.jpg)
CHA’s in the 90s+CHA’s in the 90s+Signed Compositions
Mantaci-Reutenauer ‘95
Packed Words
Hivert ‘99
Set Compositions
1
13
2
3
23
Parking Functions
Novelli-Thibon ‘04
![Page 70: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/70.jpg)
A zoo of Hopf Algberas
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Hopf’sgraded
1
1
3
2
3
2
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![Page 71: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/71.jpg)
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
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9
10
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13
A CLassification Theorem
![Page 72: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/72.jpg)
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
14
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4 5
6
7
8
9
10
11
12
13
A CLassification Theorem
Aguiar Species CHAs
![Page 73: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/73.jpg)
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
14
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4 5
6
7
8
9
10
11
12
13
A CLassification Theorem
Aguiar Species CHAs
(N)Bergeron-Lam-LiRep theory
![Page 74: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/74.jpg)
Hopf algebras of setpartitions/compositions
![Page 75: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/75.jpg)
Hopf algebras of setpartitions/compositions
![Page 76: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/76.jpg)
Hopf algebras of setpartitions/compositions
Set composition definition:
![Page 77: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/77.jpg)
Hopf algebras of setpartitions/compositions
Set composition definition:
Set partition definition:
![Page 78: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/78.jpg)
Another type of non-commutative symmetric functions
Invariants under the left actionon the polynomial ring
Free algebra generated by one element at each degree
Wolf 1936, Rosas-Sagan ’03
Invariants under the left actionon the non-comm poly ring
![Page 79: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/79.jpg)
Monomial set partition
For each monomial
Associate a set partition
![Page 80: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/80.jpg)
Example:
![Page 81: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/81.jpg)
Example:
![Page 82: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/82.jpg)
Example:
![Page 83: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/83.jpg)
Example:
This is a non commutative polynomial for a given
![Page 84: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/84.jpg)
Example:
Consider the elements to be the object by letting the
number of variables in
This is a non commutative polynomial for a given
![Page 85: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/85.jpg)
Combinatorial rule for product
![Page 86: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/86.jpg)
Combinatorial rule for product
![Page 87: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/87.jpg)
Combinatorial rule for product
![Page 88: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/88.jpg)
CoproductInspiration:
![Page 89: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/89.jpg)
CoproductInspiration:
![Page 90: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/90.jpg)
CoproductInspiration:
![Page 91: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/91.jpg)
Definition of
Non-commutativeCo-commutative
Hopf algebra of set partitions
![Page 92: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/92.jpg)
Analogy between Sym and NCSym
![Page 93: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/93.jpg)
Properties of NCSym
![Page 94: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/94.jpg)
Properties of NCSym
Non-commutative and co-commutative
![Page 95: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/95.jpg)
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)
![Page 96: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/96.jpg)
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, ...
![Page 97: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/97.jpg)
Some questions to ask
![Page 98: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/98.jpg)
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
![Page 99: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/99.jpg)
Some questions to ask
what is a fundamental basis of this space (analogue of Schur basis)?
What is the connection with representation theory?
![Page 100: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/100.jpg)
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?
![Page 101: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/101.jpg)
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
![Page 102: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/102.jpg)
The connection between NSym and NCSym
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...NCSym has graded dimensions
NSym has graded dimensions
1, 1, 2, 4, 8, 16, 32, 64, 128, ...
![Page 103: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/103.jpg)
The connection between NSym and NCSym
1, 1, 2, 5, 15, 52, 203, 877, 4140, ...NCSym has graded dimensions
NSym has graded dimensions
1, 1, 2, 4, 8, 16, 32, 64, 128, ...
There exists a Hopf morphism
![Page 104: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/104.jpg)
Families of morphisms
![Page 105: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/105.jpg)
Families of morphisms
![Page 106: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/106.jpg)
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1
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3
![Page 107: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/107.jpg)
One last open question
![Page 108: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/108.jpg)
One last open question
What is the Hopf algebra of lions?
![Page 109: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/109.jpg)
One last open question
What is the Hopf algebra of lions?
![Page 110: Non-Commutative Symmetric Functions I A zoo of Hopf algebrasgarsia.math.yorku.ca/~zabrocki/talks/HopfzooKingston.pdf · Non-Commutative Symmetric Functions I: I will present an overview](https://reader031.fdocuments.us/reader031/viewer/2022011903/5f157b09bfec0b120e65aea6/html5/thumbnails/110.jpg)
One last open question
What is the Hopf algebra of lions?
Rowr!