[Paper Review] A natural representation model for symmetric groups

Vijay Kodiyalam, Daya-Nand Verma|ArXiv.org|Feb 13, 2004

Advanced Algebra and Geometry1 references18 citations

TL;DR

This paper constructs a natural complex representation of the symmetric group $\Sigma_N$ on the vector space spanned by its involutions, showing that this representation contains each irreducible representation of $\Sigma_N$ exactly once. The construction uses a signed conjugation action on involutions, and the key result is that the endomorphism algebra of this representation is commutative and has dimension equal to the number of integer partitions of $N$, confirming multiplicity-free decomposition into all irreducibles.

ABSTRACT

For any positive integer $N$, we describe a natural complex representation of the symmetric group $Σ_N$ on the vector space spanned by its involutions that contains each irreducible representation exactly once.

Motivation & Objective

• To construct a natural complex representation of the symmetric group $\Sigma_N$ that contains each irreducible representation exactly once.
• To provide an alternative to existing Gelfand model constructions for symmetric groups, particularly those based on differential operators or Weyl algebras.
• To establish a structural explanation for the numerical coincidence between the number of involutions in $\Sigma_N$ and the sum of degrees of its irreducible representations.
• To characterize the endomorphism algebra of the representation space and show it is commutative, implying multiplicity-free decomposition.

Proposed method

• Define a representation $\pi_j$ of $\Sigma_N$ on the vector space $V_j$ spanned by $j$-fold products of disjoint transpositions (involutions of length $j$).
• Introduce a sign function $S(\sigma, \tau) = (-1)^{\#\{k : \sigma(p_{2k-1}) > \sigma(p_{2k})\}}$ to define the action of $\sigma \in \Sigma_N$ on a basis element $\tau \in X_j$.
• Show that the representation $\pi_j$ is well-defined by relating it to the $j$-th graded component of the quotient algebra $\mathrm{Sym}(\bigwedge^2 V)/I$, where $I$ is a monomial ideal.
• Prove that the endomorphism algebra $\mathrm{End}_G(V_j)$ is commutative by showing that all $G$-intertwiners are represented by symmetric matrices.
• Use orbit analysis under simultaneous conjugation on $X_j \times X_j$ to compute $\dim \mathrm{End}_G(V_j)$, showing it equals the number of partitions of $N$ with $N - 2j$ odd parts.
• Conclude that $\mathrm{End}_G(A) = \bigoplus_j \mathrm{End}_G(V_j)$ is commutative and has dimension equal to the number of partitions of $N$, hence $A$ contains all irreducible representations of $\Sigma_N$ exactly once.

Experimental results

Research questions

• RQ1 Can a natural representation of $\Sigma_N$ be constructed on the span of its involutions that realizes each irreducible representation exactly once?
• RQ2 How does the signed conjugation action on involutions give rise to a multiplicity-free representation of $\Sigma_N$?
• RQ3 What is the structure of the endomorphism algebra $\mathrm{End}_G(V_j)$ for the $j$-th involution space $V_j$?
• RQ4 Is there a structural explanation for the identity $\sum_{\text{involutions}} \dim \rho = \# \text{irreps}$, known from the RSK correspondence?
• RQ5 How do the simultaneous conjugation orbits on $X_j \times X_j$ classify the $G$-intertwiners and determine the dimension of $\mathrm{End}_G(V_j)$?

Key findings

• The representation space $A = \bigoplus_{j=0}^{\lfloor N/2\rfloor} V_j$, where $V_j$ is the span of involutions of length $j$, carries a natural $\Sigma_N$-action that contains each irreducible representation exactly once.
• The endomorphism algebra $\mathrm{End}_G(A)$ is commutative, which implies that the decomposition of $A$ into irreducible representations is multiplicity-free.
• The dimension of $\mathrm{End}_G(V_j)$ equals the number of integer partitions of $N$ with exactly $N - 2j$ odd parts.
• The $G$-intertwiners between $V_j$ and itself are represented by symmetric matrices, which follows from the sign function $S(\sigma, \tau)$ and the existence of conjugating involutions between pairs of involutions of the same length.
• The simultaneous conjugation orbits on $X_j \times X_j$ are classified by the numerical characteristic partition of the pair, which is a partition of $N$ with $N - 2j$ odd parts.
• The total dimension of $\mathrm{End}_G(A)$ is equal to the number of conjugacy classes of $\Sigma_N$, which is also the number of irreducible representations, confirming that all irreducibles appear in $A$.