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Or: How

I learned

To

Stop

Worrying

And

Love

Plethysm

SuperTableaux

GSCC 2024

Carnegie Mellon

Aditya Khanna

Virginia Tech

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Algebraic Introduction

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Representation Theoretic Motivation

Let V and W be finite dimensional vector spaces.

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Representation Theoretic Motivation

 

Let V and W be finite dimensional vector spaces.

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Representation Theoretic Motivation

 

Let V and W be finite dimensional vector spaces.

 

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Representation Theoretic Motivation

Consider two polynomial representations

 

 

 

 

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Symmetric Polynomials

A multivariate polynomial where swapping variables preserves the polynomial.

 

Examples

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Symmetric Polynomials

A multivariate polynomial where swapping variables preserves the polynomial.

 

Non-example

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Symmetric Polynomials

 

 

 

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Plethysm

 

There are many important symmetric functions.

Arguably, the most important of them all are Macdonald polynomials as they specialize to other symmetric functions.

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Plethysm

 

We will try to understand plethysm as a kind of function composition.

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Plethysm

Macdonald polynomials were defined in 1988 as a set of polynomials which satisfy certain orthogonality relations.

In 2005, Haglund, Haiman and Loehr gave a combinatorial formula for modified Macdonald polynomials using tableaux.

Macdonald polynomials can be expressed in plethystic notation quite naturally and various results can be proved by doing plethystic computations.

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Power Sum Symmetric Functions

 

 

Definition

 

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Power Sum Basis: An example

 

 

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Power Sum Basis: An example

 

 

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Power Sum Basis: An example

 

 

 

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Power Sum Symmetric Functions: But why?

 

 

 

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Universal Mapping Property (UMP)

 

 

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Plethysm

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Plethysm Axioms

 

Axiom 1

 

Axiom 2

 

Axiom 3

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Plethysm Axioms: a calculation scheme

 

Axiom 3

 

 

 

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Plethysm Axioms: a calculation scheme

 

Axiom 3

 

 

 

Axiom 2

 

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Plethysm Axioms: a calculation scheme

 

Axiom 3

 

 

 

Axiom 2

 

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Plethysm Axioms: a calculation scheme

 

 

 

Axiom 2

 

Axiom 1

 

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Plethysm Axioms: a calculation scheme

 

 

 

Axiom 2

 

Axiom 1

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Plethysm Axioms: a calculation scheme

 

and that’s it!

 

Axiom 2

 

Axiom 1

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Plethysm Alphabet

 

 

But we want to know what these different alphabets can be? Arbitrary integer coefficiented polynomials? 👀

People are always coming up with new alphabets and new things to plethysm with!

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Plethysm Alphabet

 

 

Sentiment

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Monomial Substitution Rule

 

Axiom 1

 

 

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Monomial Substitution Rule

 

Axiom 1

 

 

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Monomial Substitution Rule

But wait, we can do more!

 

This only works for MONIC polynomials!

Warning

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Monomial Substitution Rule: an example

 

 

 

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Progress!

 

But can we extend it to polynomials with integer coefficients?

 

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Negation Rule

 

 

 

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Negation Rule

 

 

 

 

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Combinatorics

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Partitions

 

 

We can represent them using Ferrer’s diagrams. For the example, we have the following picture:

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Skew Partitions

 

 

 

 

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Skew Partitions

 

 

 

 

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Skew Partitions

 

 

 

 

 

 

 

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Semistandard Young Tableaux (SSYTs)

We can fill the boxes with numbers from 1 to N…

… such that the numbers increase weakly along the rows…

…and increase strictly down the columns. For example, for N = 9:

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Monomial from an SSYT

 

For the following SSYT…

…we have the following monomial:

​

​

​

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

 

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Monomial from an SSYT

 

For the following SSYT…

​

​

T =

 

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Monomials from skew SSYTs

We can define a semi-standard tableau on a skew partition similarly and associate a monomial in the same way.

For the following SSYT…

​

​

U =

 

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Schur Polynomials and Schur Functions

Summing over all these monomials gives us the Schur polynomial corresponding to that partition.

The abstract function where we allow boxes to be filled by all natural numbers is called a Schur function.

 

Fun fact

Schur functions are symmetric functions!

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Plethysm of Schur Functions

 

 

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Algebraic Combinatorialist’s Day Job

We started with partitions, which are combinatorial objects…

…and each partition gives us a Schur polynomial which we interpret plethystically …

… and we can extend this interpretation purely algebraically…

… but does that mean anything combinatorially??

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Plethystic Schur Functions

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Plethystic Addition Formula for Schur Functions

 

 

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Plethystic Addition Formula for Schur Functions

 

But we have to be careful.

 

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Plethystic Addition Formula for Schur Functions

 

But we have to be careful.

 

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Horizontal/Vertical Strip

A skew partition is called a horizontal strip if no boxes are in the same column.

A skew partition is called a vertical strip if no boxes are in the same row.

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An SSYT is built out of horizontal strips

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What does this mean for Schur functions?

 

With the above condition and the plethystic addition formula, we can create an algebraic formulation that agrees with our combinatorial one.

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

 

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Plethystic Addition Formula for Schur Functions

 

 

 

 

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Schur negation formula

 

 

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Schur negation formula

 

 

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Conjugate of a partition

 

 

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Conjugate of a partition

 

 

 

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Enter vertical strips!

 

Variables with a positive sign will be represented by horizontal strips

Variables with a negative sign will be represented by vertical strips

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Superized

Tableaux

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Superized Tableaux

 

 

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Superized Tableaux

 

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Superized Tableaux

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

 

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Superized Tableaux

 

 

 

 

 

 

 

 

 

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All hail super tableaux!

 

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Superized Tableaux

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux

 

 

 

 

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Superized Tableaux: in the news??

arxiv: 2403.03443

Uploaded 6th March 2024

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Superized Tableaux: in the news??

arxiv: 2403.03443

Uploaded 6th March 2024

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Plethystic bargain sale!

 

 

 

 

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Parting words

Representation Theory

Motivation

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Parting words

Representation Theory

Algebra of Plethysm

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Parting words

Representation Theory

Combinatorial functions

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Parting words

Representation Theory

Plethystic Interpretation

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Parting words

Representation Theory

Putting it all together!

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Parting words

If you find a polynomial with a combinatorial interpretation, ask yourself “Is there a plethystic interpretation of this?”

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Parting words

Thank you for listening :)

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References

Loehr, N.A., Remmel, J.B. A computational and combinatorial exposé of plethystic calculus. J Algebr Comb 33, 163–198 (2011).

Macdonald I. G. Symmetric Functions and Hall Polynomials. (1979).

Alexandersson P. symmetricfunctions.com

Zabrocki M. Introduction to Symmetric Functions