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An Introduction to

Polysymmetric Functions

and the combinatorics of their transition matrices

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An Introduction to

Polysymmetric Functions

A presentation by

Aditya Khanna

(based on joint work with Nicholas Loehr)

AMS Southeastern Sectional, Tulane University

October 4, 2025

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Outline

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Outline

Symmetric Functions and Polysymmetric Functions

Origins and appearances in Geometry and Representation Theory

Transition matrix combinatorics for polysymmetric functions

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Fair warning

I might skip over some technical details in favor of the polysymmetric propaganda.

But feel free to ask me after the talk!

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Part 1

Symmetric functions

and polysymmetric functions

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Partitions

Definition

We will visualize partitions using Ferrer’s diagrams (English notation).

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Symmetric Functions

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Symmetric Functions

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This function is invariant under swapping of its variables.

Symmetric Functions

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Definition

Symmetric Functions

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Some special bases

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Some special bases

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Some special bases

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Some special bases

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Some special bases

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Now on to polysymmetric functions!

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Remember the diagrams from before? Let us fill them with numbers.

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We call this a semi-standard Young tableau (SSYT)

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As it turns out, it will be sufficient to use a partition for content as permuting the entries doesn’t change the Kostka number.

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Transition Matrices

With all this talk of bases and vector spaces, it only makes sense to talk about matrices between those bases.

We fix some ordering on the partitions. This indexes our vectors and the matrix rows and columns.

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h in s using horizontal strips

“Adding a horizontal strip” means adding boxes to a partition diagram such that no two boxes are in the same column.

SSYTs can be made by successively adding horizontal strips.

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h in s using horizontal strips

SSYTs can be made by successively adding horizontal strips.

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Interlude

Get ready for some generalized déjà vu.

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(Splitting) Types

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(Splitting) Types

In their paper, Asvin G and Andrew O’Desky introduce the idea of splitting types. This object has appeared a few times in literature before.

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degrees

(Splitting) Types

In their paper, Asvin G and Andrew O’Desky introduce the idea of splitting types. This object has appeared a few times in literature before.

multiplicities

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What do we see in the exponents?

Partitions!

So, we can associate a tensor diagram as follows:

(Splitting) Types

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Polysymmetric Functions

In symmetric functions (and polynomials), these all have degree 1

This set of doubly indexed variables has degree given by the first index.

deg = 1

deg = 3

deg = 12

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Polysymmetric Functions

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This can be thought of as making an SSYT in each factor and tensoring them together.

Pure-tensor bases

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Plethystic bases

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Plethystic bases

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Plethystic bases

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Plethystic bases

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Plethystic bases

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Plethystic bases

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So…

We talked about polysymmetric functions, but we haven’t covered anything in the title of this special session.

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Part 2

Origins and appearances in Geometry and Representation Theory

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The universal expansion of the plethystic exponential is given by

The plethystic exponential

What if we plug in a formal power series here?

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The universal expansion of the plethystic exponential is given by

The plethystic exponential

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Geometry

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Geometry

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Geometry

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G. and O’Desky mention Specht’s construction.

Representation Theory

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G. and O’Desky mention Specht’s construction.

Representation Theory

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There is a relation to the representation theory of the Uniform Block Permutation algebra [OSSZ].

Permutations can be represented using such diagrams

1 2 3 4 5 6 7 8

2 5 7 3 1 6 8 4

Representation Theory

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We can map subsets of the same size to each other – uniform blocks.

{13} {248} {57} {6}

{28} {137} {45} {6}

Representation Theory

There is a relation to the representation theory of the Uniform Block Permutation algebra [OSSZ].

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Here’s the crashiest of all courses on the representation theory of UBP algebras.

Representation Theory

There is a relation to the representation theory of the Uniform Block Permutation algebra [OSSZ].

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Representation Theory

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A wonderful coincidence

If you want to know the details

You know the drill by now…

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If you want to know the details

You know the drill by now…

A wonderful coincidence

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enough talking about things i don’t know

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Part 3

Combinatorics of transition matrices of polysymmetric functions

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What do they look like combinatorially?

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Ribbons

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Ribbons

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Horizontal Strips

The colored boxes form a horizontal strip which goes weakly upwards.

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Horizontal Strips

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This is called the SXP rule [Wildon][Shimozono, Remmel]

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This is called the SXP rule [Wildon][Shimozono, Remmel]

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This is called the SXP rule [Wildon][Shimozono, Remmel]

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Published

Preprint

(sneak peek)

Preliminary results

No direct results

Progress Table

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Transition Matrices between Plethystic Bases

Preprint coming out soon!

Contains 10+ sign-reversing involutions!

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Thank you! (btw I am on the job market)

Our paper

The SXP rule

A simple proof of the Littlewood–Richardson rule and applications (Remmel and Shimozono)
A combinatorial proof of a plethystic Murnaghan–Nakayama rule (Mark Wildon)

Polysymmetric Functions

Configuration spaces, graded spaces, and polysymmetric functions (Asvin G and Andrew O’Desky)

Uniform Block Permutation algebra

Plethysm and the algebra of uniform block permutations (Rosa Orellana, Franco Saliola, Anne Schilling, Mike Zabrocki)

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