Demystifying Plethysm using Combinatorics

by Aditya Khanna

BUGCAT 2023

Or: How

I learned

To

Stop

Worrying

And

Love

Plethysm

SuperTableaux

Scrapped Alt Title

Algebraic Introduction

Representation Theoretic Motivation

Let V and W be finite dimensional vector spaces.

Representation Theoretic Motivation

Let V and W be finite dimensional vector spaces.

Representation Theoretic Motivation

Let V and W be finite dimensional vector spaces.

Representation Theoretic Motivation

Consider two polynomial representations

Symmetric Polynomials

A multivariate polynomial where swapping variables preserves the polynomial.

Examples

Symmetric Polynomials

A multivariate polynomial where swapping variables preserves the polynomial.

Non-example

Symmetric Polynomials

Plethysm

There are many important symmetric functions.

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

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.

Power Sum Symmetric Functions

Definition

Power Sum Basis: An example

Power Sum Basis: An example

Power Sum Basis: An example

Power Sum Symmetric Functions: But why?

Universal Mapping Property (UMP)

Plethysm

Plethysm Axioms

Axiom 1

Axiom 2

Axiom 3

Plethysm Axioms: a calculation scheme

Axiom 3

Plethysm Axioms: a calculation scheme

Axiom 3

Axiom 2

Plethysm Axioms: a calculation scheme

Axiom 3

Axiom 2

Plethysm Axioms: a calculation scheme

Axiom 2

Axiom 1

Plethysm Axioms: a calculation scheme

Axiom 2

Axiom 1

Plethysm Axioms: a calculation scheme

and that’s it!

Axiom 2

Axiom 1

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!

Plethysm Alphabet

Sentiment

Monomial Substitution Rule

Axiom 1

Monomial Substitution Rule

Axiom 1

Monomial Substitution Rule

But wait, we can do more!

This only works for MONIC polynomials!

Warning

Monomial Substitution Rule: an example

Progress!

But can we extend it to polynomials with integer coefficients?

Negation Rule

Negation Rule

Combinatorics

Partitions

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

Skew Partitions

Skew Partitions

Skew Partitions

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:

Monomial from an SSYT

For the following SSYT…

…we have the following monomial:

​

​

​

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

Monomial from an SSYT

For the following SSYT…

​

​

T =

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 =

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!

Plethysm of Schur Functions

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??

Plethystic Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

But we have to be careful.

Plethystic Addition Formula for Schur Functions

But we have to be careful.

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.

An SSYT is built out of horizontal strips

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.

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Plethystic Addition Formula for Schur Functions

Schur negation formula

Schur negation formula

Conjugate of a partition

Conjugate of a partition

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

Superized

Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

All hail super tableaux!

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Superized Tableaux

Plethystic bargain sale!

Parting words

Representation Theory

Motivation

Parting words

Parting words

Parting words

Parting words

Parting words

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

Parting words

Thank you for listening :)

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