Plethysm is the algebra of composing representation-building operations. If a Schur functor S_μ first transforms a vector space V and another Schur functor S_λ is then applied to the result, the composite S_λ(S_μ(V)) can be decomposed into irreducible Schur functors. The multiplicities are plethysm coefficients.
This is harder than an ordinary tensor product. Littlewood–Richardson coefficients answer how S_λ(V)⊗S_μ(V) decomposes. Plethysm asks what happens when one representation functor is substituted into another. The operation is nonlinear in the representation ring, and no tableau rule of comparable universality is known.
The subject connects the BTT routes through Littlewood–Richardson Multiplicities, Schur–Weyl Duality, and Young Tableaux.
Schur functors · Symmetric functions · Power sums · Worked examples · Plethysm coefficients · Stable range · Practice
Tensor product and composition are different operations
Let V be a finite-dimensional complex vector space. The tensor product Sym²V⊗Sym²V is built by taking two independently constructed representations and tensoring them.
By contrast, Sym²(Sym²V) first creates the vector space Sym²V and then takes its symmetric square.
There is no reason for these two constructions to agree. In fact they have different dimensions and different irreducible decompositions.
This distinction is the first checkpoint in plethysm.
Schur functors as representation-building machines
A partition λ defines a Schur functor S_λ. Important special cases are
- S_(r)(V)=Sym^rV;
- S_(1^r)(V)=Λ^rV;
- general S_λ(V) imposes mixed row symmetries and column antisymmetries.
Given partitions λ and μ, the composite
S_λ(S_μ(V))
is again a polynomial GL(V)-representation and hence decomposes into irreducible Schur functors when dim V is sufficiently large.
Write
S_λ(S_μ(V))≈⊕_ν a_{λ,μ}^{ν} S_ν(V).
The nonnegative integers a_{λ,μ}^{ν} are plethysm coefficients.
Degree multiplies under plethysm
If S_μ is homogeneous of polynomial degree |μ| and S_λ is homogeneous of degree |λ|, then the composite has degree
|λ||μ|.
Therefore only partitions ν of size |λ||μ| can appear.
This is the plethysm analogue of the size test |ν|=|λ|+|μ| for Littlewood–Richardson coefficients.
Plethysm of symmetric functions
The representation operation is mirrored in the ring Λ of symmetric functions. If f and g are symmetric functions, their plethysm is written
f[g].
Schur functions satisfy
s_λ[s_μ]=Σ_ν a_{λ,μ}^{ν}s_ν.
The same coefficients appear in the decomposition of S_λ(S_μ(V)) in stable rank.
Plethysm is associative in the expected lambda-ring sense when carefully defined, but it is not commutative: f[g] and g[f] can differ.
Power sums make the substitution rule transparent
Let p_r=x_1^r+x_2^r+··· be the r-th power-sum symmetric function.
Plethysm is determined by the rule
p_r[g]=g(x_1^r,x_2^r,…)
together with algebra-homomorphism behavior in the outer argument.
Equivalently, on power sums p_r[p_s]=p_{rs}.
This simple multiplicative index rule is the computational backbone of symbolic plethysm.
Symmetric and exterior powers
Two of the most common plethysms are
- Sym^r(Sym^sV), corresponding to h_r[h_s] or s_(r)[s_(s)];
- Λ^r(Sym^sV), corresponding to e_r[h_s] or s_(1^r)[s_(s)].
These occur naturally in invariant theory, algebraic geometry and the study of polynomial identities.
Worked example: Sym²(Sym²V)
Assume dim V is large enough. Then
Sym²(Sym²V)≈S_(4)(V)⊕S_(2,2)(V).
At the symmetric-function level,
s_(2)[s_(2)]=s_(4)+s_(2,2).
For dim V=2, Sym²V has dimension 3, so the left side has dimension 6. On the right, S_(4)(C²)=Sym⁴(C²) has dimension 5 and S_(2,2)(C²)=det² has dimension 1. Thus 6=5+1.
This gives a complete small plethysm check.
Worked example: Λ²(Sym²V)
For sufficiently large V,
Λ²(Sym²V)≈S_(3,1)(V).
For V=C², the left side has dimension C(3,2)=3. The GL₂ highest weight (3,1) has dimension 3−1+1=3. The dimensions agree.
Combining symmetric and alternating squares therefore recovers the ordinary tensor square:
Sym²V⊗Sym²V≈S_(4)⊕S_(3,1)⊕S_(2,2).
This is also the Littlewood–Richardson decomposition of S_(2)⊗S_(2), demonstrating how tensor product and plethysm interact without being the same operation.
Worked example: Sym³(Sym²V)
A classical decomposition is
Sym³(Sym²V)≈S_(6)(V)⊕S_(4,2)(V)⊕S_(2,2,2)(V)
in stable rank.
For V=C² the last partition has three rows and vanishes. The left side has dim Sym³(C³)=C(5,3)=10. On the right, Sym⁶(C²) has dimension 7 and S_(4,2)(C²) has dimension 4−2+1=3. Thus 10=7+3.
The stable formula automatically truncates when the ambient dimension is too small.
Plethysm is not commutative
Compare Sym²(Sym³V) with Sym³(Sym²V). Both have total polynomial degree 6, but the functors differ.
For V=C², Sym³V has dimension 4, so Sym²(Sym³V) has dimension 10. Sym²V has dimension 3, so Sym³(Sym²V) also has dimension 10 in this particular case.
Equal dimensions do not imply equal representations. Their highest-weight decompositions differ in general rank.
This is another reminder that dimension checks are necessary but not sufficient.
Why plethysm coefficients are difficult
Littlewood–Richardson coefficients have several positive combinatorial rules: tableaux, hives, honeycombs, crystals and Schubert intersections.
General plethysm coefficients do not currently have one comparably simple universal positive combinatorial rule.
They can be computed through:
- power-sum expansion and character theory;
- symmetric-function algorithms;
- wreath-product representations;
- Schur functor methods;
- geometric and asymptotic techniques in special families.
The difficulty is structural, not merely computational inconvenience.
Wreath products provide a symmetric-group interpretation
Plethysm can be related to induction from wreath products S_m≀S_n inside S_{mn}.
Roughly, one symmetric group acts inside blocks while another permutes the blocks. This nested permutation symmetry mirrors composition of polynomial functors.
The exact correspondence depends on whether one uses symmetric powers, exterior powers or a general Specht-module construction, but wreath products provide the natural finite-group architecture behind plethysm.
Adams operations
The plethysm p_r[f] is often called the r-th Adams operation ψ^r(f) in lambda-ring language.
On characters, Adams operations correspond to evaluating a representation character on r-th powers of group elements under suitable compact-group interpretations:
ψ^r(χ)(g)=χ(g^r).
This is another bridge between symmetric-function substitution and representation-theoretic operations.
Plethystic notation in generating functions
Modern algebraic combinatorics often uses plethystic notation more broadly than literal Schur-functor composition. An “alphabet” X can be treated symbolically, and expressions such as f[X+Y], f[XY] or Ω[X] compactly encode operations on symmetric functions.
This notation is powerful but convention-sensitive. In particular, constants and minus signs inside plethystic brackets do not always behave like ordinary variable substitution.
When using plethystic notation, state whether the context is lambda-ring substitution, alphabet notation or literal polynomial substitution.
Foulkes-type problems
Foulkes’ conjecture compares Sym^a(Sym^bV) and Sym^b(Sym^aV) when a≤b. One formulation predicts an inclusion of characters in stable rank, equivalently inequalities among certain plethysm coefficients.
The problem remains open in general and illustrates how difficult even apparently symmetric plethysm families can be.
Therefore one should not assume that swapping the two symmetric-power indices yields an obvious dominance relation.
Hermite reciprocity in rank two
For two-dimensional V, a remarkable special identity states
Sym^m(Sym^nV)≈Sym^n(Sym^mV)
as SL₂ representations.
This explains why some low-rank dimension and decomposition comparisons appear unexpectedly symmetric.
Hermite reciprocity is special to rank two and should not be promoted to a general commutativity theorem for plethysm.
Stable rank
A Schur functor S_ν(V) vanishes if the number of rows ℓ(ν) exceeds dim V.
Thus a plethysm identity in the abstract symmetric-function ring represents the stable decomposition. Specializing to V=C^n deletes terms with too many rows.
This is analogous to Schur–Weyl duality and Deligne categories: the stable combinatorial formula exists independently of one small ambient rank, then finite-rank constraints remove inaccessible components.
Connection to algebraic complexity
Plethysm coefficients appear in geometric complexity theory, where one studies coordinate rings of orbit closures associated with polynomial computation models.
Representation multiplicities can serve as potential obstructions separating orbit closures. This links plethysm to questions inspired by computational complexity such as permanent versus determinant.
The existence of a representation-theoretic obstruction does not by itself resolve a complexity separation; one must prove the required multiplicity behavior and geometric implication.
A verification workflow
- 1. Distinguish tensor product from functor composition.
- 2. Check degree: |ν|=|λ||μ|.
- 3. Translate Schur functors to Schur functions.
- 4. Use power sums for symbolic substitution.
- 5. State ambient rank.
- 6. Remove partitions with too many rows after stable decomposition.
- 7. Verify dimensions in small rank.
- 8. Do not infer equality from equal dimensions.
- 9. Distinguish plethysm coefficients from LR and Kronecker coefficients.
- 10. Treat broad positivity conjectures as conjectures unless proved for the chosen family.
Common mistakes
- Writing s_λs_μ when the intended operation is s_λ[s_μ].
- Assuming plethysm is commutative.
- Using LR tableaux to compute arbitrary plethysm coefficients.
- Forgetting the multiplicative degree test.
- Keeping stable summands that vanish in small rank.
- Confusing power-sum substitution with ordinary scalar substitution.
- Assuming Hermite reciprocity in every rank.
- Treating open Foulkes-type statements as established theorems.
Practice questions
1. What is the polynomial degree of S_(3)(S_(2)(V))? 2. State the symmetric-function plethysm notation. 3. What is p_3[p_2]? 4. Decompose Sym²(Sym²V) in stable rank.
5. Decompose Λ²(Sym²V). 6. Verify the first decomposition for V=C² by dimension. 7. What is the key difference between LR and plethysm coefficients? 8. What finite groups appear naturally behind plethysm?
9. What is an Adams operation? 10. What happens to a stable summand S_ν(C^n) when ℓ(ν)>n? 11. What special reciprocity holds for two-dimensional V? 12. Why is a dimension check not enough?
Worked answers
1. 3·2=6.
2. s_λ[s_μ].
3. p_6.
4. S_(4)⊕S_(2,2).
5. S_(3,1).
6. Sym²(C³) has dimension 6; Sym⁴(C²) has dimension 5 and det² has dimension 1.
7. LR coefficients decompose tensor products; plethysm coefficients decompose compositions of Schur functors.
8. Wreath products.
9. The operation ψ^r(f)=p_r[f], corresponding in suitable character contexts to g↦g^r.
10. It vanishes.
11. Hermite reciprocity Sym^m(Sym^nV)≈Sym^n(Sym^mV) as SL₂ representations.
12. Nonisomorphic representations can have equal total dimension.
Sources and further study
Standard references include I. G. Macdonald, Symmetric Functions and Hall Polynomials; Richard Stanley, Enumerative Combinatorics, Vol. 2; and Fulton–Harris, Representation Theory. Modern work on plethysm connects to symmetric functions, wreath products, geometric complexity theory and asymptotic representation theory. The BTT Littlewood–Richardson guide provides the closest comparison.
Representation Mathematics — Batch 10
Interpolate stable rank through Deligne Categories. Study paired commuting symmetries in Howe Duality. Move to symplectic geometry in The Orbit Method. Return to the BTT Mathematics Learning Hub.