Plethysm

2026-10-15 Improve this page

#Plethysm

See [Hag07] and [LR10] for an introduction to calculations with plethysm. Another good reference is M. Zabrocki’s introduction to symmetric functions [Zab03, Ch. 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 $f.$

R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki study plethysm through the uniform block permutation algebra, also called the party algebra [OSSZ22]. They define a Frobenius characteristic map whose character theory naturally produces plethysms of Schur functions.

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

#Definition

Plethysm is closely related to the power-sum symmetric functions. The following properties uniquely define plethysm, 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 $f[g].$

The operator sending $f(\xvec)$ to $f(\xvec^k) = f[\powerSum_k(\xvec)]$ is called the $k$-th Adams operator. The adjoint to the Adams operator is the Verschiebung operator $V_k,$ which is defined as $V_k(\completeH_r) = \completeH_{r/k},$ if $k \mid r,$ and $0$ otherwise.

Example

Let us compute $\powerSum_{22}[\powerSum_{43}]$: \[\begin{aligned} \powerSum_{22}[\powerSum_{43}] &= (\powerSum_{2}[\powerSum_{43}])^2 \\ &= (\powerSum_{2}[\powerSum_{4}] \cdot \powerSum_{2}[\powerSum_{3}])^2 \\ &= (\powerSum_{8} \cdot \powerSum_{6})^2 \\ &= \powerSum_{8866}. \end{aligned}\]

Similarly, $\powerSum_{32}[3\powerSum_{4}]$ can be computed as \[\begin{aligned} \powerSum_{32}[3\powerSum_{4}] &= (\powerSum_{3}[\powerSum_{4}+\powerSum_{4}+\powerSum_{4}]) (\powerSum_{2}[\powerSum_{4}+\powerSum_{4}+\powerSum_{4}]) \\ &= (3\powerSum_{3}[\powerSum_{4}])(3\powerSum_{2}[\powerSum_{4}]) \\ &= 9\powerSum_{12} \cdot \powerSum_{8}. \end{aligned}\]

The situation is a bit more involved if the coefficients of $f$ and $g$ in the plethysm $f[g]$ 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

For example, \[\powerSum_{k}[5q \cdot \powerSum_{m}] = 5 q^k \powerSum_{km}.\]

Capital letters are sometimes used to denote the sum of the variables in that alphabet. For example, $X = x_1+x_2+\dotsb = \elementaryE_1(\xvec).$

A. D’Andrea, E. Fatighenti, and C. Onorati study a derivation $\mathsf{D}$ on $\spaceSym$ and prove a chain rule for $\mathsf{D}$ with respect to plethysm [DFO25]. They also relate the Schur-support geometry of $\mathsf{D}(f)$ to that of a homogeneous symmetric function $f.$

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

The last identity generalizes as follows: \[\schurS_\nu[f\cdot g] = \sum_{\lambda,\mu} g^{\nu}_{\lambda\mu} \schurS_\lambda[f] \schurS_\mu[g].\]

We 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

#Bosonic plethysm and cluster algebras

The bosonic plethysm coefficients are the multiplicities \[\schurS_\mu[\schurS_2] = \sum_\lambda a^\lambda_{\mu,(2)}\schurS_\lambda,\] equivalently the multiplicities of $\mathbb{S}_\lambda V$ in $\mathbb{S}_\mu(\operatorname{Sym}^2V).$

Let $V=\setC^\ell,$ $W=\setC^m,$ and let $U_V$ be a maximal unipotent subgroup of $\GL(V).$ Y. Fan and J. Fei identify the highest-weight algebra \[R_{\ell,m} = \operatorname{Sym}(\operatorname{Sym}^2V\otimes W)^{U_V}\] with an upper cluster algebra $\mathcal{U}(\Sigma_{\ell,m})$ having polynomial frozen coefficients [FF26]. The seed is skew-symmetrizable and admits a reddening sequence, so theta functions extending across the frozen boundary form a basis indexed by lattice points in a rational polyhedral cone $C_{\ell,m}.$

For a weak composition $\alpha=(\alpha_1,\dotsc,\alpha_m),$ put \[b^\lambda_{\alpha,(2)} = \dim\operatorname{Hom}_{\GL(V)} \left( \mathbb{S}_\lambda V, \bigotimes_{r=1}^m \operatorname{Sym}^{\alpha_r}(\operatorname{Sym}^2V) \right).\] These nonnegative multiplicities are lattice-point counts in weight fibers of $C_{\ell,m}.$ If $\alpha^\sigma(\mu)_i=\mu_i-i+\sigma(i),$ then Jacobi–Trudi gives \[a^\lambda_{\mu,(2)} = \sum_{\sigma\in\symS_m} \operatorname{sgn}(\sigma) b^\lambda_{\alpha^\sigma(\mu),(2)}.\] Thus symmetric-square plethysm is an explicit finite alternating sum of polyhedral lattice-point counts. The construction does not yet give a positive combinatorial rule for $a^\lambda_{\mu,(2)}$ itself.

Proposition

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

Milo Bechtloff Weising proves a wreath-product generalization of Littlewood reciprocity [Wei25]. The result gives branching multiplicities for irreducible $G^n\rtimes \symS_n$-representations inside highest-weight $\mathrm{GL}_{nm}(\setC)$ representations, extending the classical Littlewood rule for the embedded symmetric group.

The SXP rule gives a formula for the plethysm [Wil18]: \[\schurS_\lambda[\powerSum_r] = \sum_{\nuvec} \sign_r(\nuvec^\ast) c^{\lambda}_{\nuvec} \schurS_{\nuvec^\ast}.\] See also the plethystic Murnaghan–Nakayama rule. W. F. Doran gives a tableau formula for $\powerSum_\mu[\schurS_{(a)}]$ using roots of unity and a major-index-like statistic on semistandard tableaux [Dor98]. I. M. Gessel and Y. Zhuang give plethystic formulas for permutation enumeration [GZ20].

Florence Maas-Gariépy and É. Tétreault develop a tableau approach to plethysms $\schurS_\nu[\completeH_\lambda]$ and $\schurS_\nu[\elementaryE_\lambda]$ [MT22]. They use ribbon tableaux, lattice points in polytopes, and a Kronecker map to analyze the case where $\nu$ is a partition of $3$ and then extend to arbitrary $\lambda.$

Á. Gutiérrez and M. H. Rosas prove partial symmetries for large families of iterated plethysm coefficients [GR23]. Their flip symmetry generalizes the usual $\omega$-involution symmetry and is preserved by certain plethystic operations. Á. Gutiérrez, R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki study bivariate generating functions for plethysm coefficients with bounded output length [GOSS+25]. They prove rationality, give an explicit $q$-Ehrhart-theoretic algorithm in the two-row case, and derive linear recursions for $\mathrm{SL}_2$ plethysm coefficients. C. Bessenrodt, C. Bowman, and R. Paget classify multiplicity-free plethysms of Schur functions [BBP22]. R. Paget and M. Wildon study plethysms through representations of $\mathrm{SL}_2(\setC)$ [PW21]. They classify several families of plethystic isomorphisms, including cases with conjugate partitions, rectangles, and partitions with at most two rows or columns. I. Pak, G. Panova, and J. P. Swanson give a positive combinatorial interpretation for plethysm coefficients $\langle \schurS_\mu[\schurS_\nu],\schurS_\lambda\rangle$ when $\lambda$ has at most two rows [PPS25]. Their model counts explicit marked trees and, in the case $\mu=(n),$ also interprets corresponding rectangular Kronecker coefficients.

Proposition (See [BN21, Appendix B]).

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

Moreover, D. Yeliussizov shows that the plethysm is in fact a specialization of the canonical stable Grothendieck polynomial [Yel17]: \[\grothendieckStable^{(c,-c)}_\lambda(\xvec) = \schurS_\lambda\left[ \frac{\xvec}{1- c \xvec} \right].\]

#The s-perp trick

In [COSS+24], 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. They give algorithms for monomial and Schur expansions of symmetric functions, and in several special cases these are faster than the corresponding SageMath routines.

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 $f=g.$ The 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

For example, \[\schurS_{211}[\schurS_{11}] = \schurS_{3 3 2} + \schurS_{3 2 2 1} + \schurS_{4 2 1 1} + \schurS_{2 2 2 1 1} + \schurS_{3 2 1 1 1} + \schurS_{3 1 1 1 1 1}.\]

The Schur plethysm coefficients stabilize under some operations. In [PW25] Paget and Wildon prove a general theorem, which specializes to all known stability results regarding plethysm coefficients. For example, the main stability result of [LO23] is proved.

For a partition $\lambda,$ set $\lambda[mn]=(mn-|\lambda|,\lambda_1,\lambda_2,\dotsc).$ S. Sarkar proves that $\langle \completeH_n[\completeH_m],\schurS_{\lambda[mn]}\rangle$ is constant for $m\geq\lambda_1$ and $n\geq\lfloor(|\lambda|+\lambda_1)/2\rfloor,$ and identifies its stable value as $\langle \schurS_\lambda,H[H-1-\completeH_1]\rangle,$ where $H=\sum_{i\geq0}\completeH_i$ [Sar26]. For $\lambda_1\leq4,$ the paper also gives a positive formula for this value as the number of lattice points in an explicit polytope.

#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. É. Tétreault introduces a relation on integer partitions motivated by Schur positivity of differences of plethysms [Te20]. The restriction to one-part partitions is equivalent to the classical Foulkes conjecture, and the paper proves several properties via explicit inclusions of modules.

H. Zhu studies plethysm through orbit harmonics of loci of unordered set partitions [Zhu26]. Applying orbit harmonics to $\Pi_{(2^a)}$ and $\Pi_{(a^2)}$ gives graded $\symS_{2a}$-modules whose graded characters refine the Schur expansions of $\completeH_a[\completeH_2]$ and $\completeH_2[\completeH_a].$ The paper also formulates a Foulkes-type conjecture from the pair $\Pi_{(b^a)},$ $\Pi_{(a^b)}$ and proves the case $b=2.$

Example (A small orbit-harmonics refinement).

For $a=3,$ Zhu’s graded Frobenius formulas [Zhu26, Thms. 3.3 and 3.5] give \[\operatorname{grFrob}(R(\Pi_{(2^3)});q) = \schurS_{6}+q\schurS_{42}+q^2\schurS_{222}\] and \[\operatorname{grFrob}(R(\Pi_{(3^2)});q) = \schurS_{6}+q\schurS_{42}.\] After forgetting the grading by setting $q=1,$ these become the plethysms \[\completeH_3[\completeH_2] = \schurS_{6}+\schurS_{42}+\schurS_{222} \quad\text{and}\quad \completeH_2[\completeH_3] = \schurS_{6}+\schurS_{42}.\] Thus in this first nontrivial case of Foulkes’ conjecture, the difference is the single Schur-positive term \[\completeH_3[\completeH_2]-\completeH_2[\completeH_3] = \schurS_{222}.\]

Bibliography

[AU20]Per Alexandersson and Joakim Uhlin. Cyclic sieving, skew Macdonald polynomials and Schur positivity. Algebraic Combinatorics, 3(4):913–939, 2020.

.bib

@article{AlexanderssonUhlin2020,
     author = {Alexandersson, Per and Uhlin, Joakim},
     title = {Cyclic sieving, skew {M}acdonald polynomials and {S}chur positivity},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {4},
     year = {2020},
     pages = {913-939},
     doi = {10.5802/alco.123},
     language = {en},
     url2 = {alco.centre-mersenne.org/item/ALCO_2020__3_4_913_0/}
}

[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.

.bib

@article{BentoNovaes2021,
  doi = {10.1088/1751-8121/abcf58},
  url2 = {https://doi.org/10.1088/1751-8121/abcf58},
  year = {2021},
  month = mar,
  publisher = {{IOP} Publishing},
  volume = {54},
  number = {12},
  pages = {125201},
  author = {Pedro H S Bento and Marcel Novaes},
  title = {Semiclassical treatment of quantum chaotic transport with a tunnel barrier},
  journal = {Journal of Physics A: Mathematical and Theoretical}
}

[BBP22]Christine Bessenrodt, Chris Bowman and Rowena Paget. The classification of multiplicity-free plethysms of Schur functions. Transactions of the American Mathematical Society, 375(7):5151–5194, 2022.

.bib

@article{BessenrodtBowmanPaget2022,
  author = {Christine Bessenrodt and Chris Bowman and Rowena Paget},
  title = {The classification of multiplicity-free plethysms of {S}chur
    functions},
  year = {2022},
  journal = {Transactions of the American Mathematical Society},
  volume = {375},
  number = {7},
  pages = {5151--5194},
  doi = {10.1090/tran/8642},
  eprint = {2001.08763}
}

[COSS+24]Laura Colmenarejo, Rosa Orellana, Franco Saliola, Anne Schilling and Mike Zabrocki. The mystery of plethysm coefficients. Open problems in algebraic combinatorics, 110:275–292, 2024.

.bib

@incollection{ColmenarejoOrellanaSaliolaSchillingZabrocki2022x,
  author = {Laura Colmenarejo and Rosa Orellana and Franco Saliola and
    Anne Schilling and Mike Zabrocki},
  title = {The mystery of plethysm coefficients},
  year = {2024},
  booktitle = {Open Problems in Algebraic Combinatorics},
  series = {Proceedings of Symposia in Pure Mathematics},
  volume = {110},
  pages = {275--292},
  publisher = {American Mathematical Society},
  doi = {10.1090/pspum/110},
  url = {https://doi.org/10.1090/pspum/110},
  eprint = {2208.07258}
}

[DFO25]Alessandro D’Andrea, Enrico Fatighenti and Claudio Onorati. A plethystic chain rule. arXiv:2508.03568, 2025.

.bib

@article{DAndreaFatighentiOnorati2025x,
  author = {Alessandro D'Andrea and Enrico Fatighenti and Claudio Onorati},
  title = {A plethystic chain rule},
  year = {2025},
  eprint = {2508.03568},
  url = {https://arxiv.org/abs/2508.03568},
  journal = {arXiv e-prints}
}

[Dor98]William F. Doran. A New Plethysm Formula for Symmetric Functions. Journal of Algebraic Combinatorics, 8(3):253–272, 1998.

.bib

@article{Doran1998,
  author = {William F. Doran},
  title = {A {N}ew {P}lethysm {F}ormula for {S}ymmetric {F}unctions},
  year = {1998},
  journal = {Journal of Algebraic Combinatorics},
  volume = {8},
  number = {3},
  pages = {253--272},
  doi = {10.1023/A:1008662732304}
}

[FF26]Yelin Fan and Jiarui Fei. Cluster Algebras for Bosonic Plethysm. arXiv:2608.00963v1, 2026.

.bib

@article{FanFei2026x,
  author = {Yelin Fan and Jiarui Fei},
  title = {Cluster {A}lgebras for {B}osonic {P}lethysm},
  year = {2026},
  eprint = {2608.00963v1},
  url = {https://arxiv.org/abs/2608.00963v1},
  journal = {arXiv e-prints}
}

[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.

.bib

@article{Foulkes1950,
  doi = {10.1112/jlms/s1-25.3.205},
  url2 = {https://doi.org/10.1112/jlms/s1-25.3.205},
  year = {1950},
  month = jul,
  publisher = {Wiley},
  volume = {s1-25},
  number = {3},
  pages = {205--209},
  author = {H. O. Foulkes},
  title = {Concomitants of the Quintic and Sextic Up To Degree Four in the Coefficients of the Ground Form},
  journal = {Journal of the London Mathematical Society}
}

[GV85]Ira Gessel and Gérard Viennot. Binomial determinants, paths, and hook length formulae. Advances in Mathematics, 58(3):300–321, December 1985.

.bib

@article{GesselViennot1985,
  doi = {10.1016/0001-8708(85)90121-5},
  url2 = {https://doi.org/10.1016/0001-8708(85)90121-5},
  year = {1985},
  month = dec,
  publisher = {Elsevier {BV}},
  volume = {58},
  number = {3},
  pages = {300--321},
  author = {Ira Gessel and G{\'{e}}rard Viennot},
  title = {Binomial determinants,  paths, and hook length formulae},
  journal = {Advances in Mathematics}
}

[GZ20]Ira M. Gessel and Yan Zhuang. Plethystic formulas for permutation enumeration. Advances in Mathematics, 375:107370, 2020.

.bib

@article{GesselZhuang2020,
  author = {Ira M. Gessel and Yan Zhuang},
  title = {Plethystic formulas for permutation enumeration},
  year = {2020},
  journal = {Advances in Mathematics},
  volume = {375},
  pages = {107370},
  doi = {10.1016/j.aim.2020.107370},
  eprint = {2001.00654}
}

[GOSS+25]Gutiérrez, Rosa Orellana, Franco Saliola, Anne Schilling and Mike Zabrocki. A geometric and generating function approach to plethysm. arXiv:2511.02649, 2025.

.bib

@article{GutierrezOrellanaSaliolaSchillingZabrocki2025x,
  author = {{\'A}lvaro Guti{\'e}rrez and Rosa Orellana and Franco Saliola and
    Anne Schilling and Mike Zabrocki},
  title = {A geometric and generating function approach to plethysm},
  year = {2025},
  eprint = {2511.02649},
  url = {https://arxiv.org/abs/2511.02649},
  journal = {arXiv e-prints}
}

[GR23]Gutiérrez and Mercedes H. Rosas. Partial symmetries of iterated plethysms. Annals of Combinatorics, 27(3):493–518, 2023.

.bib

@article{GutierrezRosas2023,
  author = {{\'A}lvaro Guti{\'e}rrez and Mercedes H. Rosas},
  title = {Partial symmetries of iterated plethysms},
  year = {2023},
  journal = {Annals of Combinatorics},
  volume = {27},
  number = {3},
  pages = {493--518},
  doi = {10.1007/s00026-023-00652-4},
  url = {https://doi.org/10.1007/s00026-023-00652-4},
  eprint = {2201.00240}
}

[LO23]Stacey Law and Yuji Okitani. Some stable plethysms. Proceedings of the American Mathematical Society, 151(11):4557–4564, August 2023.

.bib

@article{LawOkitani2023,
  title = {Some stable plethysms},
  volume = {151},
  ISSN = {1088-6826},
  url = {http://dx.doi.org/10.1090/proc/16556},
  DOI = {10.1090/proc/16556},
  number = {11},
  journal = {Proceedings of the American Mathematical Society},
  publisher = {American Mathematical Society (AMS)},
  author = {Law,  Stacey and Okitani,  Yuji},
  year = {2023},
  month = aug,
  pages = {4557–-4564}
}

[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.

.bib

@article{LoehrRemmel2010,
  doi = {10.1007/s10801-010-0238-4},
  url2 = {https://doi.org/10.1007/s10801-010-0238-4},
  year  = {2010},
  month = jun,
  publisher = {Springer Nature},
  volume = {33},
  number = {2},
  pages = {163--198},
  author = {Nicholas A. Loehr and Jeffrey B. Remmel},
  title = {A computational and combinatorial expos{\'{e}} of plethystic calculus},
  journal = {Journal of Algebraic Combinatorics}
}

[MT22]Florence Maas-Gariépy and Étienne Tétreault. 3-Plethysms of homogeneous and elementary symmetric functions. arXiv:2209.14942, 2022.

.bib

@article{MaasGariepyTetreault2022x,
  author = {Florence Maas-Gariépy and Étienne Tétreault},
  title = {3-{P}lethysms of homogeneous and elementary symmetric functions},
  year = {2022},
  eprint = {2209.14942},
  url = {https://arxiv.org/abs/2209.14942},
  journal = {arXiv e-prints}
}

[OSSZ22]Rosa Orellana, Franco Saliola, Anne Schilling and Mike Zabrocki. Plethysm and the algebra of uniform block permutations. Algebraic Combinatorics, 5(5):1165–1203, 2022.

.bib

@article{OrellanaSaliolaSchillingZabrocki2022Party,
  author = {Rosa Orellana and Franco Saliola and Anne Schilling and
    Mike Zabrocki},
  title = {Plethysm and the algebra of uniform block permutations},
  year = {2022},
  journal = {Algebraic Combinatorics},
  volume = {5},
  number = {5},
  pages = {1165--1203},
  doi = {10.5802/alco.243},
  url = {https://doi.org/10.5802/alco.243},
  eprint = {2112.13909}
}

[PW21]Rowena Paget and Mark Wildon. Plethysms of symmetric functions and representations of ${\mathrm{SL}}_2(\mathbb C)$. Algebraic Combinatorics, 4(1):27–68, 2021.

.bib

@article{PagetWildon2021,
  author = {Rowena Paget and Mark Wildon},
  title = {Plethysms of symmetric functions and representations of
    {${\mathrm{SL}}_2(\mathbb C)$}},
  year = {2021},
  journal = {Algebraic Combinatorics},
  volume = {4},
  number = {1},
  pages = {27--68},
  doi = {10.5802/alco.150}
}

[PW25]Rowena Paget and Mark Wildon. Two stability theorems on plethysms of Schur functions. arXiv:2505.06104, 2025.

.bib

@article{PagetWildon2025,
  Author  = {Rowena Paget and Mark Wildon},
  Title   = {Two Stability Theorems on Plethysms of {S}chur Functions},
  Year    = {2025},
  Eprint  = {2505.06104},
  url = {https://arxiv.org/abs/2505.06104},
  Journal = {arXiv e-prints}
}

[PPS25]Igor Pak, Greta Panova and Joshua P. Swanson. A combinatorial interpretation for certain plethysm and Kronecker coefficients. arXiv:2511.02312, 2025.

.bib

@article{PakPanovaSwanson2025x,
  author = {Igor Pak and Greta Panova and Joshua P. Swanson},
  title = {A combinatorial interpretation for certain plethysm and
    {K}ronecker coefficients},
  year = {2025},
  eprint = {2511.02312},
  url = {https://arxiv.org/abs/2511.02312},
  journal = {arXiv e-prints}
}

[Sar26]Soumyadip Sarkar. Stability of plethysm coefficients and modified polynomial induction. arXiv:2610.07929v2, 2026.

.bib

@article{Sarkar2026x,
  author = {Soumyadip Sarkar},
  title = {Stability of plethysm coefficients and modified polynomial induction},
  year = {2026},
  eprint = {2610.07929v2},
  url = {https://arxiv.org/abs/2610.07929v2},
  journal = {arXiv e-prints}
}

[Te20]Tétreault. A new order on integer partitions. arXiv:2010.15333, 2020.

.bib

@article{Tetreault2020PartitionOrder,
  author = {{\'E}tienne T{\'e}treault},
  title = {A new order on integer partitions},
  year = {2020},
  eprint = {2010.15333},
  url = {https://arxiv.org/abs/2010.15333},
  journal = {arXiv e-prints}
}

[Wei25]Milo Bechtloff Weising. Wreath generalization of Littlewood reciprocity. arXiv:2506.07727, 2025.

.bib

@article{Weising2025Littlewoodx,
  author = {Milo Bechtloff Weising},
  title = {Wreath Generalization of {L}ittlewood Reciprocity},
  year = {2025},
  eprint = {2506.07727},
  url = {https://arxiv.org/abs/2506.07727},
  journal = {arXiv e-prints}
}

[Wil18]Mark Wildon. A generalized SXP rule proved by bijections and involutions. Annals of Combinatorics, 22(4):885–905, November 2018.

.bib

@article{Wildon2018,
  doi = {10.1007/s00026-018-0409-x},
  url2 = {https://doi.org/10.1007/s00026-018-0409-x},
  year = {2018},
  month = nov,
  publisher = {Springer Nature},
  volume = {22},
  number = {4},
  pages = {885--905},
  author = {Mark Wildon},
  title = {A Generalized {SXP} Rule Proved by Bijections and Involutions},
  journal = {Annals of Combinatorics}
}

[Yel17]Damir Yeliussizov. Duality and deformations of stable Grothendieck polynomials. Journal of Algebraic Combinatorics, 45(1):295–344, September 2017.

.bib

@article{Yeliussizov2017,
  doi = {10.1007/s10801-016-0708-4},
  url2 = {https://doi.org/10.1007/s10801-016-0708-4},
  year = {2017},
  month = sep,
  publisher = {Springer Science and Business Media {LLC}},
  volume = {45},
  number = {1},
  pages = {295--344},
  author = {Damir Yeliussizov},
  title = {Duality and deformations of stable {G}rothendieck polynomials},
  journal = {Journal of Algebraic Combinatorics}
}

[Zab03]Mike Zabrocki. Introduction to symmetric functions. Lecture notes, York University, 2003. Chapter 3

.bib

@misc{Zabrocki2003,
  title        = {Introduction to Symmetric Functions},
  author       = {Mike Zabrocki},
  howpublished = {Lecture notes, York University},
  year         = {2003},
  note         = {Chapter 3},
  url          = {https://garsia.math.yorku.ca/ghana03/chapters/mainfile3.pdf}
}

[Zhu26]Hai Zhu. Plethysm and orbit harmonics. arXiv:2602.12623, 2026.

.bib

@article{Zhu2026x,
  author = {Hai Zhu},
  title = {Plethysm and orbit harmonics},
  year = {2026},
  eprint = {2602.12623},
  url = {https://arxiv.org/abs/2602.12623},
  journal = {arXiv e-prints}
}

[Hag07]James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.

.bib

@book{qtCatalanBook,
  Author = {James Haglund},
  Title = {The $q,t$-{C}atalan numbers and the space of diagonal harmonics ({U}niversity lecture series)},
  Publisher = {American Mathematical Society},
  Year = {2007},
  ISBN = {0821844113},
  url = {https://www.math.upenn.edu/~jhaglund/books/qtcat.pdf}
}