2023-09-22

## Plethysm

See [Hag07] and [LR10] for an introduction to calculations with plethysm. Another good reference is Mike Zabrocki's introduction to symmetric functions, Chapter III.

### Representation-theoretic interpretation

Plethysm can easily be described using representation theory of $\GL_n.$

Let $U,V,W$ be vector spaces over $\setC,$ and consider polynomial representations $\phi : \GL(U) \to \GL(V)$ and $\psi : \GL(V) \to \GL(W).$ Let $f$ and $g$ be the corresponding characters (symmetric functions). The composition $\psi\circ \phi : \GL(U) \to \GL(W)$ is then also a polynomial representation, and $g[f]$ is the plethysm of $g$ and $f.$

This can be made more concrete, and extended to any symmetric function, see below.

### Definition

Plethysm is closely related to to the power-sum symmetric functions.
The following properties uniquely define pletysm where $f,$ $g$ and $h$ denote symmetric functions
with *integer* coefficients. Brackets are commonly used to denote plethystic substitutions.

- $\powerSum_k[\powerSum_m] = \powerSum_{km}.$
- $\powerSum_k[f\pm g] = \powerSum_k[f] \pm \powerSum_k[g].$
- $\powerSum_k[f\cdot g] = \powerSum_k[f] \cdot \powerSum_k[g].$
- $(f\pm g)[h] = f[h] \pm g[h].$
- $(f\cdot g)[h] = f[h] \cdot g[h].$

Since every symmetric function can be expressed as a sum of products of $\powerSum_k(\xvec),$ these properties uniquely determine the expression $f[g].$

**Example.**

Let us compute $\powerSum_{22}[\powerSum_{43}].$ We have that

\begin{align*} \powerSum_{22}[\powerSum_{43}] &= (\powerSum_{2}[\powerSum_{43}])^2 \\ &= (\powerSum_{2}[\powerSum_{4}] \cdot \powerSum_{2}[\powerSum_{3}])^2 \\ &= (\powerSum_{8} \cdot \powerSum_{6})^2 \\ &= \powerSum_{8866}. \end{align*}Similarly, $\powerSum_{32}[3\powerSum_{4}]$ can be computed as

\begin{align*} \powerSum_{32}[3\powerSum_{4}] &= (\powerSum_{3}[\powerSum_{4}+\powerSum_{4}+\powerSum_{4}])(\powerSum_{2}[\powerSum_{4}+\powerSum_{4}+\powerSum_{4}]) \\ &= (3\powerSum_{3}[\powerSum_{4}])(3\powerSum_{2}[\powerSum_{4}]) \\ &= 9\powerSum_{12} \cdot \powerSum_{8}. \end{align*}Warning! The situation is a bit more involved if the coefficients of $f$ and $g$ in the pletysm $f[g]$ are formal power series in say $\setC(\qvec).$

We first define the plethysm $\powerSum_k[g(\xvec)].$ If $g(\xvec) = \sum_{\mu} d_\mu(\qvec) \powerSum_\mu(\xvec)$ then

\[ \powerSum_k[g(\xvec)] \coloneqq \sum_{\mu} d_\mu(q_1^k,q_2^k,\dotsc) \; \powerSum_{k\mu}(\xvec) . \]Let now $f(\xvec) = \sum_{\lambda} c_\lambda(\qvec) \powerSum_\lambda(\xvec).$ Then

\[ f[g(\xvec)] \coloneqq \sum_{\lambda} c_\lambda(\qvec) \prod_{j=1}^{\length(\lambda)} \powerSum_{\lambda_j}[g(\xvec)]. \]This definition agrees with the previous one in the case $f$ and $g$ are symmetric functions with integer coefficients, not depending on formal parameters.

**Example.**

We have that

\[ \powerSum_{k}[5q \cdot \powerSum_{m}] = 5 q^k \powerSum_{km}. \]Capital letters are sometimes used to denote the sum of the variables in that alphabet. For example, $X = x_1+x_2+\dotsb = \elementaryE_1(\xvec).$

### Identities

We have the following identities:

- $\powerSum_{k}[qX] = q^k\powerSum_{k}[X]$
- $\powerSum_{k}[-X] = -\powerSum_{k}[X]$
- $\powerSum_{k}[\epsilon X] = \epsilon^k \powerSum_{k}[X]$ when $\epsilon=-1.$
- $\powerSum_{k}[X(1-q)] = (1-q^k)\powerSum_{k}[X]$
- $\powerSum_{k}[X/(1-q)] = \powerSum_{k}[X]/(1-q^k)$
- $\powerSum_{\lambda}[X+Y] = \powerSum_{\lambda}[X]+\powerSum_{\lambda}[Y]$
- $\powerSum_{\lambda}[XY] = \powerSum_{\lambda}[X]\powerSum_{\lambda}[Y]$
- $\schurS_{\lambda}[X+Y] = \sum_{\mu \subseteq \lambda} \schurS_{\mu}[X]\schurS_{\lambda/\mu}[Y]$
- $\schurS_\nu[XY] = \sum_{\lambda,\mu} g^{\nu}_{\lambda\mu} \schurS_\lambda(\xvec) \schurS_\mu(\yvec)$ where $g^{\nu}_{\lambda\mu}$ are the Kronecker coefficients.

The last identity generalizes as follows:

\[ \schurS_\nu[f\cdot g] = \sum_{\lambda,\mu} g^{\nu}_{\lambda\mu} \schurS_\lambda[f] \schurS_\mu[g]. \]I have not found a reference for the following useful relation but it is easy to prove. It is used in [AU20].

**Proposition (Alexandersson, 2019).**

Let $f$ be a homogeneous symmetric function of degree $n.$ Then

\[ \powerSum_k[ \omega f ] = (-1)^{n(k+1)}\omega(\powerSum_k[ f ]). \]### Other identities

**Proposition.**

We have that

\[ \completeH_k[\completeH_2] = \sum_{\mu : \text{ even }} \schurS_{\mu} \]where the sum ranges over all partitions of $2k$ into even parts. This identity is due to Littlewood.

The SXP rule (see [Wil18] for a generalization), gives a formula for the plethysm,

\[ \schurS_\lambda[\powerSum_r] = \sum_{\nuvec} \sign_r(\nuvec^\ast) c^{\lambda}_{\nuvec} \schurS_{\nuvec^\ast}. \]See also the plethystic Murnaghan–Nakayama rule.

**Proposition (See [Appendix B, BN21]).**

We have that for an indeterminate $c,$

\[ \schurS_\lambda\left[ \frac{\xvec}{1- c \xvec} \right] = \sum_{\rho} c^{|\rho|-|\lambda|} \det\left[ \binom{ \rho_j-j }{\mu_i-i} \right] \cdot \schurS_\rho(\xvec). \]The determinant of binomial coefficients have a combinatorial interpretation, see [GV85].

Moreover, it is shown in [Yel17] that the plethysm is in fact a specialization of the canonical stable Grothenieck polynomial:

\[ \grothendieckStable^{(c,-c)}_\lambda(\xvec) = \schurS_\lambda\left[ \frac{\xvec}{1- c \xvec} \right]. \]## The s-perp trick

In [COSSZ22], the authors describe in detail how to use the so-called s-perp trick in order to compute certain plethysm coefficients. This trick can also be used in many other settings.

First, define the following operator on a symmetric function $f$:

\[ \schurS_\lambda^{\perp} f \coloneqq \sum_{\mu} \langle f , \schurS_\lambda \schurS_\mu \rangle \schurS_\mu. \]**Proposition.**

Let $f$ and $g$ be two homogeneous symmetric functions of degree $d.$ Then

\[ \schurS_r^{\perp} f = \schurS_r^{\perp} g \text{ for all $1\leq r \leq d$} \]implies that $f=g.$ Same holds if we replace $\schurS_r = \completeH_r$ with $\schurS_{1^r} = \elementaryE_r.$

## Open problems

### Schur plethysm

**Problem (See [LR10] for more info).**

Let $\lambda \vdash n$ and $\mu \vdash m.$ Find a formula for the Schur plethysm coefficients in

\[ \schurS_\lambda[\schurS_\mu] = \sum_{\nu \vdash mn} p^{\nu}_{\lambda \mu} \schurS_{\nu}. \]This is a major open problem in the theory of symmetric functions and representation theory of classical groups.

**Example.**

We have that

\[ \schurS_{211}[\schurS_{11}] = \schurS_{3 3 2} + \schurS_{3 2 2 1} + \schurS_{4 2 1 1} + \schurS_{2 2 2 1 1} + \schurS_{3 2 1 1 1} + \schurS_{3 1 1 1 1 1}. \]### Foulkes conjecture

Foulkes conjecture [Fou50] states that $\completeH_b[\completeH_a]-\completeH_a[\completeH_b]$ is Schur-positive whenever $a \leq b.$ See the cycle index polynomial page for more information.

## References

- [AU20] Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913-939, 2020.
- [BN21] Pedro H S Bento and Marcel Novaes. Semiclassical treatment of quantum chaotic transport with a tunnel barrier. Journal of Physics A: Mathematical and Theoretical, 54(12):125201, March 2021.
- [COSSZ22] Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling and Mike Zabrocki. The mystery of plethysm coefficients. arXiv e-prints, 2022.
- [Fou50] H. O. Foulkes. Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form. Journal of the London Mathematical Society, s1-25(3):205–209, July 1950.
- [GV85] Ira Gessel and Gérard Viennot. Binomial determinants, paths, and hook length formulae. Advances in Mathematics, 58(3):300–321, December 1985.
- [Hag07] James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.
- [LR10] Nicholas A. Loehr and Jeffrey B. Remmel. A computational and combinatorial exposé of plethystic calculus. Journal of Algebraic Combinatorics, 33(2):163–198, June 2010.
- [Wil18] Mark Wildon. A generalized SXP rule proved by bijections and involutions. Annals of Combinatorics, 22(4):885–905, November 2018.
- [Yel17] Damir Yeliussizov. Duality and deformations of stable Grothendieck polynomials. Journal of Algebraic Combinatorics, 45(1):295–344, September 2017.