Saxl's Conjecture in Representation Theory

Updated 20 December 2025

Saxl's Conjecture is a key statement in asymptotic representation theory asserting that the tensor square of the staircase partition’s irreducible representation contains every irreducible of the symmetric group.
The conjecture was proven unconditionally through a blend of dominance order analysis, modular representation theory, and detailed combinatorial partition techniques.
Its resolution provides actionable insights into Kronecker coefficients, the structure of 2-regular partitions, and offers promising avenues for extensions in finite group and Lie theory.

Saxl’s Conjecture is a principal statement in the asymptotic representation theory of the symmetric group and Kronecker coefficients. It predicts maximal universality in the constituent structure of the tensor square of the irreducible associated to the staircase partition. Its recent unconditional proof resolves a central problem in algebraic combinatorics and sheds new light on the interplay between dominance order, modular representation theory, and the combinatorics of partitions.

1. Formulation and Background

Saxl’s Conjecture, formulated by Jan Saxl in 2012, asserts the following: for the staircase partition $\rho_k = (k, k-1, \dots, 1)$ of the $k$ th triangular number $T_k = k(k+1)/2$ , the square of the corresponding irreducible representation of the symmetric group $S_{T_k}$ contains every irreducible module as a constituent. Formally,

$g(\lambda, \rho_k, \rho_k) \ge 1 \quad \text{for all } \lambda \vdash T_k,$

where $g(\lambda, \mu, \nu)$ is the Kronecker coefficient, i.e., the multiplicity of $S^\lambda$ in $S^\mu \otimes S^\nu$ for complex $S_n$ -modules. This conjecture is motivated by the question of how large the support of a tensor square can be, and whether a canonical choice— $\rho_k$ —suffices to generate the entire space of irreducibles as constituents (Lee, 17 Dec 2025).

2. Algebraic and Combinatorial Framework

The proof of Saxl’s Conjecture rests on several key notions:

Staircase partition: $\rho_k = (k, k-1, ..., 1)$ ; size $T_k$ .
Dominance order: For partitions $\lambda, \mu \vdash n$ , $\lambda \succeq \mu$ if and only if $\sum_{i=1}^j \lambda_i \ge \sum_{i=1}^j \mu_i$ for all $j$ .
2-regular partition: A partition with all distinct parts. The set of 2-regular partitions of $n$ is $R_n$ .
Decomposition matrix (characteristic 2): $D = (d_{\lambda\mu})$ , where $d_{\lambda\mu}$ is the multiplicity of the irreducible $D^\mu$ in the reduction modulo 2 of $S^\lambda$ .

These objects interact through the modular representation theory of symmetric groups, the combinatorics of partitions, and block theory.

3. Proof Strategy and Structural Theorems

The unconditional proof (Lee, 17 Dec 2025) employs the following sequence of structural results:

Staircase Minimality Theorem: Among all 2-regular partitions of $T_k$ , $\rho_k$ is uniquely minimal in the dominance order. Thus, for all $\mu\in R_{T_k}$ , $\mu \succeq \rho_k$ with equality iff $\mu = \rho_k$ .
Ikenmeyer's Kronecker Positivity: For partitions $\lambda, \rho_k \vdash T_k$ , if $\lambda$ is dominance-comparable to $\rho_k$ , then $g(\lambda, \rho_k, \rho_k) \ge 1$ . As every 2-regular $\mu$ satisfies $\mu \succeq \rho_k$ , this establishes positivity for all 2-regular constituents.
Modular Saturation and Lifting: Employing James’s theorem on the decomposition matrix in characteristic 2, one has $d_{\mu\mu} = 1$ for all $\mu\in R_n$ , and $d_{\lambda\mu} > 0 \implies \lambda \succeq \mu$ . Defining modular saturation—the property that the projective square in characteristic 2 contains all 2-regular constituents—one shows that $\rho_k$ achieves this property.
Bessenrodt–Bowman–Sutton Lifting Theorem: If a 2-core $\gamma$ achieves modular saturation in characteristic 2, then $g(\lambda,\gamma,\gamma) > 0$ for every partition $\lambda\vdash n$ . Thus, every ordinary partition occurs in the tensor square over $\mathbb{C}$ .
Characterization of Kronecker-Universal Partitions: For $n = T_k$ , the equivalence between $g(\lambda,\mu,\mu) > 0$ for all $\lambda$ , $\mu = \rho_k$ , $\mu$ a 2-core, and $S^\mu$ projective modulo 2 shows that the staircase is the unique self-conjugate Kronecker-universal partition at triangular numbers.

Prior to the unconditional proof, significant partial progress was made via:

Dominance Criteria and Hooks: Work of Ikenmeyer, Pak–Panova–Vallejo, and Bessenrodt established positivity for all hooks, two-row partitions, and many double-hook shapes via a combination of character evaluations and modular theory (Pak et al., 2013, Bessenrodt, 2017). Pak–Panova–Vallejo introduced a sufficient condition for positivity, reducing $g(\lambda, \rho_k, \rho_k)>0$ to the non-vanishing of a character value on a special conjugacy class (principal hooks) and applying the Murnaghan–Nakayama rule and unimodality arguments.
Semi-Group Property and Telescopic Constructions: Papers such as (Ebrahimi, 5 Nov 2025, Luo et al., 2015) demonstrated that Kronecker coefficients are compatible with the addition of partitions, producing large telescopic families of guaranteed constituents using Manivel's semi-group property and the block theory of the symmetric group.
2-Modular Filtrations and 2-Separated Partitions: Techniques leveraging the semisimplicity of Specht modules labeled by 2-separated partitions yielded wide families of positive Saxl coefficients by exploiting the structure of decomposition matrices and the linkage to projective covers in characteristic 2 (Bessenrodt et al., 2019).
Higher Tensor Powers and Asymptotics: Results on the tensor cube and fourth power of the staircase module showed that, for $k \ge 3$ , the triple and quadruple tensor products contain every irreducible constituent for $S_{T_k}$ (Harman et al., 2022, Luo et al., 2015), even where the tensor square proof was previously unavailable.

5. Generalizations and Connections in Finite Group Theory

Saxl’s Conjecture has multiple group-theoretical and Lie-theoretical analogues:

Saxl Graphs: In permutation group theory, a parallel conjecture postulates that for primitive groups $G$ with base size 2, the associated Saxl graph is connected of diameter at most 2. This has been verified for all almost simple primitive groups with soluble point stabilizers (Burness et al., 2021).
Lie-Theoretic Generalizations: There is a uniform framework using the Weyl group of $A_{n-1}$ and spin representations of finite Coxeter groups (Chen et al., 2024). The conjecture extends to claiming that the tensor square of an explicit sum over the cuspidal family in Lusztig's partition contains every irreducible character, with verification for the exceptional and non-crystallographic types.
Spin-Saxl and Mixed Squares: Spin analogues of Saxl's conjecture for the double covers of $S_n$ and $A_n$ arise by analyzing squares of spin characters, with partial results indicating that a large fraction of irreducibles are achieved this way, motivating ongoing research (Bessenrodt, 2017).

6. Significance, Applications, and Open Directions

The resolution of Saxl’s Conjecture (Lee, 17 Dec 2025) represents a culmination of advances across combinatorics, symmetric function theory, modular representation theory, and computational algebra. Its significance stems from:

The explicit identification of universality in tensor squares for a natural class of partitions.
The confluence of combinatorial, character-theoretic, and modular approaches in the proof.
The clarification of when unique Kronecker-universal behavior occurs (only for staircases at triangular numbers among self-conjugate partitions).

Open directions include:

Determining refined combinatorial rules for the multiplicities in the tensor square decomposition.
Extending the structural understanding to spin covers, complex reflection groups, and affine Weyl groups.
Exploring analogues of modular saturation and universality at small characteristics or in the context of other Lie types.
Categorification and connections to diagonal harmonics and geometric complexity theory.

The proof strategy and modular arguments developed for Saxl’s Conjecture are expected to yield further insight into Kronecker coefficients, Specht module decompositions, and broader tensor product phenomena in the representation theory of finite and algebraic groups.