Young–Jucys–Murphy Elements in Symmetric Groups

• Young–Jucys–Murphy elements are a sequence of commuting elements defined as sums of transpositions that generate the Gelfand–Tsetlin subalgebra in symmetric group algebras.
• They link combinatorial tableau statistics with eigenvalue spectra, providing a foundation for constructing Murphy cellular and seminormal bases in representation theory.
• Their explicit polynomial identities and symmetric function expansions enable efficient computation of central elements and combinatorial invariants in group actions.

The Young–Jucys–Murphy (YJM) elements are a distinguished sequence of commuting elements in the group algebras of symmetric groups, pivotal to the algebraic, combinatorial, and representation-theoretic structure of these algebras. Defined as explicit sums of transpositions, these elements generate the Gelfand–Tsetlin subalgebra and provide a unifying framework for the description of seminormal forms, labeling of irreducible representations via tableau combinatorics, and the construction of canonical (cellular) bases such as the Murphy and seminormal bases. Their spectrum captures core combinatorial statistics—often “contents” of tableau boxes—and their algebraic symmetries translate into far-reaching results, including centrality of symmetric function evaluations, spectral theorems, and representation-theoretic multiplicity-freeness. The YJM paradigm extends to specialized and deformed settings (Hecke, Brauer, Sergeev superalgebras, etc.), proving to be a central organizing device in modern algebra.

1. Definition and Fundamental Properties

Given a commutative ring $\mathbf{k}$ and the group algebra $\mathbf{k}[S_n]$, the Young–Jucys–Murphy elements are defined by

$$m_1 = 0,\quad m_k = t_{1,k} + t_{2,k} + \cdots + t_{k-1, k}\text{ for }2 \leq k \leq n,$$

where $t_{i, k}$ denotes the transposition $(i\, k)$. Each $m_k$ is a $\mathbf{k}$-linear combination of transpositions and lives in $\mathbf{k}[S_n]$ (Grinberg, 28 Jul 2025).

Their central algebraic feature is commutativity: $m_i m_j = m_j m_i\quad \text{for all } i, j.$ Thus, $\mathbf{k}[m_1,\dots,m_n]$ forms a commutative subalgebra, classically known as the Gelfand–Tsetlin subalgebra (Grinberg, 28 Jul 2025).

The YJM elements satisfy several fundamental polynomial identities, notably: \begin{align*} &\prod_{i=-(k-1)}^{k-1} (m_k - i) = 0 \end{align*} for each $k$, implying that their spectrum, on $\mathbf{k}[S_n]$-modules, is contained in $\{-(k-1), \dots, k-1\}$ (Grinberg, 28 Jul 2025).

A crucial combinatorial identity is the symmetrizer product formula: $$(1 + m_1)(1 + m_2) \dots (1 + m_n) = \sum_{w \in S_n} w,$$ establishing a link between the YJM elements and the structure of the group algebra (Grinberg, 28 Jul 2025).

2. Commutativity, Centrality, and the Gelfand–Tsetlin Algebra

All YJM elements $m_1, \dots, m_n$ commute, and thus their polynomial algebra serves as the maximal commutative subalgebra in $\mathbf{k}[S_n]$ generated by the conjugacy-invariant spectral data (“contents”) associated to tableau fillings.

A fundamental centrality property holds: for any symmetric function $f$ in $n$ variables, $f(m_1,\dots,m_n) \in Z(\mathbf{k}[S_n])$, the center of the group algebra (Loth et al., 13 Mar 2024). In fact, every central element can be written as $f(m_1,\dots,m_n)$ for a suitable symmetric function $f$.

The elementary symmetric functions of the YJM elements have explicit expansions: $$e_k(m_1,\dots,m_n) = \sum_{\substack{w\in S_n\ rl(w) = k}} w,$$ where $rl(w)$ is the reflection length of $w$ (number of transpositions needed to represent $w$) (Grinberg, 28 Jul 2025). These expansions provide a transparent combinatorial correspondence between symmetric functions of the YJM elements and reflection statistics of permutations.

3. Connection to Young Tableaux and Representation Theory

Within the construction of irreducible representations (Specht modules) of $S_n$, the Gelfand–Tsetlin subalgebra acts semisimply, and the joint spectrum of $(m_1,\dots,m_n)$ on these modules encodes combinatorial information about standard Young tableaux.

For a standard tableau $t$ and a box filled with $k$ in row $i$ and column $j$, the eigenvalue of $m_k$ is $j-i$ (the “content” of that box) (Grinberg, 28 Jul 2025). Thus, in any seminormal (Murphy) basis, each $m_k$ acts diagonally with eigenvalues given by the tableau content function.

This labelling by contents (or “spectral data”) is a complete invariant: distinct standard tableaux correspond to different joint eigenvalues, and hence to different basis vectors in irreducible representations.

The YJM elements thus provide the algebraic backbone for explicit forms of the seminormal and cellular bases in the symmetric group algebra, and underlie the explicit construction and distinction of irreducible representations (Grinberg, 28 Jul 2025).

4. Key Identities and Polynomial Relations

Several pivotal algebraic identities involving the YJM elements facilitate both computational results and theoretical development:

• The symmetrizer formula: $(1 + m_1)\cdots(1 + m_n) = \sum_{w\in S_n} w$.
• The minimal polynomial: for each $k$,

$\prod_{i=-(k-1)}^{k-1} (m_k - i) = 0.$

• The symmetric function expansion: elementary symmetric polynomials $e_k(m_1,\dots,m_n)$ correspond to the sum over permutations of reflection length $k$ (Grinberg, 28 Jul 2025).

These identities serve as the cornerstone for analyzing the center, understanding representation content, and effecting the translation from algebraic to combinatorial representation-theoretic data.

5. Murphy Cellular Bases and Seminormal Forms

The connection between YJM elements and the Murphy basis is foundational. The so-called Murphy operators $(m_1,\dots,m_n)$ are used to construct the Murphy cellular basis for $\mathbf{k}[S_n]$, a basis adapted to the cellular (tabular) structure of the algebra.

Each cellular basis vector is essentially indexed by standard tableaux, with the action of the YJM elements being upper-triangular and, on suitable quotients, diagonal with eigenvalues given by the tableau content. The granular control this gives over module structure enables a clean description of the behavior of Specht modules, duals, and branching rules (Grinberg, 28 Jul 2025).

This explicit upper-triangular (or diagonal) action is directly responsible for the standard (Gelfand–Tsetlin) multiplicity-freeness in the restriction of representations from $S_n$ to $S_{n-1}$, and for the rapid computation of hook-lengths, dimensions, and the shape of the center (Grinberg, 28 Jul 2025).

6. Combinatorial and Algorithmic Applications

The YJM framework enables efficient algorithmic coding of permutations and computations in the group algebra. Techniques such as “transposition coding”—expanding permutations as products of transpositions guided by tableau data—allow for concrete combinatorial calculations and the verification of identities by elementary methods.

For instance, exercises demonstrate that simple transpositions $s_i$ act on $m_j$ as: $$s_i m_j = m_j s_i \text{ for } j\neq i,i+1,\quad s_i m_i + 1 = m_{i+1} s_i,\quad s_i m_{i+1} - 1 = m_i s_i,$$ facilitating recursive computations and clarifying the interaction of the full symmetric group algebra with the commuting YJM subalgebra (Grinberg, 28 Jul 2025).

This concrete computational accessibility underpins both the combinatorics of permutation statistics (such as nonstarters, cycles, and reflection lengths) and the algebraic structure necessary for the paper of group actions and modules.

7. Summary and Impact

The Young–Jucys–Murphy elements form the backbone of the modern combinatorial approach to the symmetric group algebra $\mathbf{k}[S_n]$. Their explicit definitions and minimal polynomial relations provide tractable algebraic tools; their commutativity generates the Gelfand–Tsetlin subalgebra, foundational for multiplicity-free branching, seminormal forms, and the construction of (cellular) module bases. The identification of their spectrum with tableau contents bridges group algebra actions and the combinatorics of Young tableaux. Moreover, symmetric polynomials in the YJM elements span the entire center of $\mathbf{k}[S_n]$, affording a uniformly explicit description of central elements.

The YJM elements are indispensable for the elementary, computational, and representation-theoretic analysis of symmetric group algebras, with influence reaching far into algebraic combinatorics, Schur–Weyl duality, and categorification programs. Their paper provides a template for understanding analogous constructions in many families of diagram algebras and quantum deformations.