10.4 Plethysm and Symmetric Group Characters

Plethysm is a powerful operation on symmetric functions that combines them in complex ways. It's like a supercharged multiplication that creates new symmetric functions from existing ones. This operation is crucial for understanding advanced topics in algebraic combinatorics.

Plethysm connects to Schur functions and tableaux combinatorics by providing a way to compose and decompose these functions. It's used to solve problems in representation theory, particularly for symmetric groups and general linear groups. Understanding plethysm helps unlock deeper insights into character theory and tensor products.

Plethysm of Symmetric Functions

Definition and Properties

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• The plethysm operation, denoted by ∘ or square brackets, is a binary operation on symmetric functions that takes two symmetric functions $f$ and $g$ and produces a new symmetric function $f[g]$
• Plethysm is a ring homomorphism from the ring of symmetric functions to itself, meaning it satisfies the properties:
  • $(f + g)[h] = f[h] + g[h]$
  • $(fg)[h] = f[h]g[h]$
• The plethysm operation is associative, i.e., $(f[g])[h] = f[g[h]]$, but not commutative, i.e., $f[g] ≠ g[f]$ in general (e.g., $s_{(2)}[s_{(1,1)}] ≠ s_{(1,1)}[s_{(2)}]$)
• The power sum symmetric functions satisfy the simplifying relation $p_n[p_m] = p_{nm}$, which can be used to compute plethysms involving power sums (e.g., $p_2[p_3] = p_6$)

Plethysm of Complete Homogeneous and Elementary Symmetric Functions

• The plethysm of complete homogeneous symmetric functions satisfies the formula $h_n[h_m] = h_{(n-1)m+1}$ and $h_n[e_m] = e_{nm}$, where $e_n$ denotes the elementary symmetric functions
  • For example, $h_3[h_2] = h_5$ and $h_2[e_3] = e_6$
• The plethysm of elementary symmetric functions satisfies the formula $e_n[h_m] = h_{nm}$ and $e_n[e_m] = e_{(n-1)m+1}$
  • For example, $e_2[h_3] = h_6$ and $e_3[e_2] = e_5$
• These formulas can be derived using the generating function approach and the properties of plethysm as a ring homomorphism

Computing Plethysm of Schur Functions

Littlewood-Richardson Rule

• The Littlewood-Richardson rule is an algorithm for computing the coefficients $c^k_{ij}$ in the product of Schur functions $s_i ∙ s_j = ∑_k c^k_{ij}s_k$, which can be adapted to compute plethysms of Schur functions
• To compute $s_λ[s_μ]$ using the Littlewood-Richardson rule:
  1. Replace each box in the Young diagram of $λ$ with the Young diagram of $μ$
  2. Linearize the resulting "diagram of diagrams" to obtain a sum of ordinary Young diagrams
  3. The resulting sum gives the expansion of $s_λ[s_μ]$ in terms of Schur functions
• Example: To compute $s_{(2)}[s_{(1,1)}]$, replace each box in the Young diagram of $(2)$ with the Young diagram of $(1,1)$, linearize, and express the result as a sum of Schur functions: $s_{(2)}[s_{(1,1)}] = s_{(2,2)} + s_{(3,1)}$

Other Methods for Computing Plethysm

• The Frobenius characteristic map $ch: R(S_n) → Λ_n$ is a ring isomorphism between the representation ring of the symmetric group $S_n$ and the ring of symmetric functions $Λ_n$ of degree $n$, which can be used to translate plethysm problems from symmetric functions to representations of $S_n$
• The Giambelli identity expresses a Schur function $s_λ$ as a determinant of hook Schur functions, which can sometimes simplify the computation of plethysms
  • For example, $s_{(3,2)} = \begin{vmatrix} s_{(3)} & s_{(4)} \\ s_{(1)} & s_{(2)} \end{vmatrix} = s_{(3)}s_{(2)} - s_{(4)}s_{(1)}$
• The Jacobi-Trudi identity expresses a Schur function $s_λ$ as a determinant of complete homogeneous symmetric functions, which can be used in conjunction with the plethysm formulas for $h_n$ to compute plethysms of Schur functions
  • For example, $s_{(2,1)} = \begin{vmatrix} h_2 & h_3 \\ 1 & h_1 \end{vmatrix} = h_2h_1 - h_3$

Schur Functions and Plethysm in Representation Theory

Symmetric Group Characters and Schur Functions

• The irreducible characters of the symmetric group $S_n$ are in bijection with the partitions of $n$, and the character values on conjugacy classes are given by the Frobenius formula $χ^λ(μ) = (|C_μ|/n!)⟨p_μ, s_λ⟩$, where $C_μ$ is the conjugacy class of permutations of cycle type $μ$
• The Frobenius characteristic map $ch$ sends an irreducible representation $V_λ$ of $S_n$ to the Schur function $s_λ$, and the character $χ^λ$ to the power sum symmetric function $p_λ$, providing a link between symmetric group characters and symmetric functions
  • For example, the irreducible representation $V_{(3,1)}$ of $S_4$ corresponds to the Schur function $s_{(3,1)}$ under the characteristic map

Plethysm and Wreath Products

• The plethysm of Schur functions corresponds to the tensor product of certain representations of the symmetric group, called wreath products, under the characteristic map
• The wreath product of symmetric groups $S_n ≀ S_m$ is defined as the semidirect product of $S_m$ with the direct product of $m$ copies of $S_n$, and its irreducible representations are indexed by partitions of $nm$
• The decomposition of the plethysm $h_n[V_λ]$ into irreducible representations of $S_m$ corresponds to the decomposition of the induced representation $Ind^{S_{nm}}_{S_n≀S_m}(V_λ ⊗ ⋯ ⊗ V_λ)$ into irreducibles, where $V_λ$ is an irreducible representation of $S_n$

Murnaghan-Nakayama Rule

• The Murnaghan-Nakayama rule is a combinatorial formula for computing the character values of symmetric group representations, which can be used in conjunction with the characteristic map to study plethysms of symmetric functions
• The rule states that the character value $χ^λ(μ)$ can be computed by summing over all rim hook tableaux of shape $λ$ and weight $μ$, with each tableau contributing a sign determined by the height of its rim hooks
• Example: To compute $χ^{(3,2,1)}(2,2,1,1)$ using the Murnaghan-Nakayama rule, sum over all rim hook tableaux of shape $(3,2,1)$ and weight $(2,2,1,1)$, with appropriate signs

Applications of Plethysm in Representation Theory

Schur Functors and GL(V)-Modules

• The plethysm $s_λ[s_μ]$ corresponds to the composition of Schur functors $S_λ ∘ S_μ$ applied to a vector space $V$, where $S_λ$ and $S_μ$ are the Schur functors associated with the partitions $λ$ and $μ$, respectively
• This provides a connection between plethysm and the representation theory of the general linear group $GL(V)$, as the Schur functors yield irreducible polynomial representations of $GL(V)$
• The decomposition of the plethysm $s_λ[s_μ]$ into Schur functions corresponds to the decomposition of the $GL(V)$-module $(S_λ ∘ S_μ)(V)$ into irreducible submodules

Cauchy Identity and Symmetric Algebra

• The Cauchy identity $∑_λ s_λ(x)s_λ(y) = ∏_{i,j}(1 - x_iy_j)^{-1}$ can be interpreted as a statement about the decomposition of the symmetric algebra $S(V ⊗ W)$ as a $GL(V) × GL(W)$-module, where $s_λ(x)$ and $s_λ(y)$ are interpreted as characters of $GL(V)$ and $GL(W)$, respectively
• The symmetric algebra $S(V)$ can be decomposed as a direct sum of irreducible $GL(V)$-modules, with the Schur module $S_λ(V)$ appearing with multiplicity equal to the dimension of the irreducible representation of the symmetric group $S_{|λ|}$ indexed by $λ$

Tensor Products and Plethysm

• The plethysm operation can be used to study the decomposition of certain tensor products of representations of the symmetric and general linear groups
• For example, the decomposition of $S^k(S^l(V))$ into irreducible $GL(V)$-modules can be obtained by computing the plethysm $h_k[h_l]$ and interpreting the result in terms of Schur functors
• Similarly, the decomposition of the tensor product $V_λ ⊗ V_μ$ of two irreducible representations of the symmetric group $S_n$ can be studied using the plethysm of Schur functions $s_λ[s_μ]$ and the characteristic map