Kronecker-Universal Self-Conjugate Partitions

• Kronecker-universal self-conjugate partitions are integer partitions that remain invariant under conjugation and whose tensor square contains every irreducible symmetric group representation.
• They are uniquely characterized by staircase partitions at triangular number sizes, where the partition ρₖ demonstrates complete universality.
• The analysis employs dominance order, rim-hook combinatorics, and character criteria to deepen understanding of symmetric group representations and Kronecker coefficients.

A Kronecker-universal self-conjugate partition is a partition $\mu\vdash n$ of an integer $n$ that is invariant under conjugation and whose tensor square $S^\mu\otimes S^\mu$ contains every irreducible constituent of $S_n$. These partitions are central in the context of symmetric group representation theory, Kronecker coefficients, and the Saxl conjecture.

1. Foundational Definitions

A partition $\lambda\vdash n$ is Kronecker-universal if for every other partition $\nu\vdash n$, the Kronecker coefficient $g(\nu,\lambda,\lambda)>0$; equivalently, $S^\lambda\otimes S^\lambda$ contains all irreducible $S_n$-representations as constituents.

A partition is self-conjugate if its Ferrers diagram is invariant under reflection across the main diagonal—that is, $\lambda' = \lambda$. Self-conjugates correspond to $S_n$-modules whose structure is particularly symmetric and are often parametrized by their principal-hook partitions.

2. Characterization Theorem for Triangular Numbers

At sizes equal to the $k$th triangular number $T_k = \frac{k(k+1)}{2}$, the main classification result is:

$\begin{theorem}[Characterization of Kronecker-universal self-conjugates at triangular numbers] Let %%%%13%%%% for %%%%14%%%%, and let %%%%15%%%% be self-conjugate. Then %%%%16%%%% is Kronecker-universal (i.e., %%%%17%%%% for all %%%%18%%%%) if and only if %%%%19%%%%. \end{theorem}$

Thus, for each triangular number size, the unique Kronecker-universal self-conjugate partition is the staircase partition $\rho_k$ (Lee, 17 Dec 2025).

3. Structural and Proof Outline

The proof consists of five conceptual steps:

1. Staircase Minimality in Dominance Order: $\rho_k$ is the unique dominance-minimal element among all 2-regular partitions of $T_k$. For $\mu\in R_{T_k}$ (strictly decreasing parts), $\mu\trianglerighteq\rho_k$, with equality only for $\mu=\rho_k$.
2. Ikenmeyer’s Dominance Criterion: For staircases $\rho_k$, if $\lambda\trianglerighteq\rho_k$ or $\rho_k\trianglerighteq\lambda$, then $g(\lambda,\rho_k,\rho_k)\ge1$. By minimality, every 2-regular partition $\mu$ satisfies $g(\mu,\rho_k,\rho_k)\ge1$.
3. Coverage of All 2-Regular Partitions: Every 2-regular partition appears in the tensor square $S^{\rho_k}\otimes S^{\rho_k}$.
4. Modular Saturation: In characteristic 2, diagonal entries $d_{\mu\mu}=1$ force modular saturation—every projective indecomposable labeled by a 2-regular partition appears in the tensor square.
5. Lifting via Bessenrodt–Bowman–Sutton: For $\rho_k$ (a 2-core), modular saturation in char 2 lifts to char 0 Kronecker positivity, so $g(\lambda, \rho_k, \rho_k)>0$ for all $\lambda\vdash T_k$.

The uniqueness among self-conjugates follows because: any Kronecker-universal self-conjugate must be a 2-core, and the only self-conjugate 2-core is the staircase (Lee, 17 Dec 2025).

4. Connections to Character Criteria and Other Shapes

Pak–Panova–Vallejo (Pak et al., 2013) introduced a character criterion: for self-conjugate $\mu$, if for some $\lambda$ the character value $\chi^\lambda[\widehat{\mu}]\neq0$ on principal-hook cycle type, then $g(\lambda,\mu,\mu)>0$. This is especially effective for staircases, with $\widehat{\rho}_k=(2k-1,2k-5,2k-9,\ldots)$.

For large $k$, this criterion, together with deep unimodality results for partitions in arithmetic progression, establishes positivity of Kronecker coefficients for all hook-shapes and all two-row shapes in the tensor square of $\rho_k$. These families, while not all irreducibles, constitute significant subclasses.

For certain other self-conjugates such as chopped-square or caret shapes, analogous character criteria apply and yield positivity for large families, but full Kronecker-universality is not settled; obstructions arise from combinatorial congruence conditions and small exceptional partitions (Pak et al., 2013, Zhao, 2023).

5. Extent of Universality and Near-Universal Shapes

Full universality at triangular sizes is exclusive to staircases. For square-shaped self-conjugate partitions $(m^m)$ of $n=m^2$, extensive positivity results exist:

• For $m\ge7$, every two-row and three-row partition except a finite set of near-hooks appears in the tensor square $S^{(m^m)}\otimes S^{(m^m)}$.
• Remaining exceptional zeros are identified explicitly as $(m^2-3,2,1)$, $(m^2-4,3,1)$, $(m^2-j,1^j)$ for $j=1,2,4,6$ (and conjugates).
• For general three-row shapes with smallest part at least 2, all appear (except small near-hook gaps).
• For large near-hook shapes with second row at least 8 (and $m\ge20$), Kronecker coefficients are positive (Zhao, 2023).

A plausible implication is that large square shapes are "nearly" Kronecker-universal for partitions of length up to 3 and near-hooks, but true universality fails due to identified finite obstructions. Full universality would require additional positivity for these exceptional shapes.

6. Virtual Characters and Further Positivity Constructions

Li (Li, 2017) extended techniques for generating nonzero Kronecker coefficients via virtual characters (rim-hook wrap-operators) applied to self-conjugate partitions. If $\chi^\lambda(\mu)\neq0$, certain nearby partitions $\tau$ with related rim-hook structure also have $\chi^\tau(\mu)\neq0$, thus $g(\lambda,\lambda,\mu)>0$.

This suggests broad families of positive Kronecker coefficients for self-conjugates and raises the possibility—still conjectural—that suitably constructed infinite sequences of self-conjugates could realize all irreducibles via virtual characters.

7. Illustrative Examples for Small $k$

The characterization is confirmed by small instances:

• $k=2$, $T_2=3$: Self-conjugates are $(2,1)$ and $(1,1,1)$; only $(2,1)$ (the staircase) is Kronecker-universal.
• $k=3$, $T_3=6$: Candidates are $(3,2,1)$ (staircase) and $(2,2,2)$; $(2,2,2)$ fails, only $(3,2,1)$ is universal.
• $k=4$, $T_4=10$: Only $(4,3,2,1)$ (staircase) is Kronecker-universal; other self-conjugates such as $(3,3,3,1)$ are not 2-cores and fail universality (Lee, 17 Dec 2025).

Summary Table: Kronecker-Universality at Triangular Sizes

Triangular Size $T_k$	Self-Conjugates $\mu\vdash T_k$	Kronecker-Universal?
3	(2,1), (1,1,1)	Only (2,1)
6	(3,2,1), (2,2,2)	Only (3,2,1)
10	(4,3,2,1), others	Only (4,3,2,1)

Only the staircase partitions $\rho_k = (k, k-1, ..., 1)$ are Kronecker-universal self-conjugates at triangular numbers.

Outlook and Open Directions

The uniqueness of staircases as Kronecker-universal self-conjugate partitions at triangular numbers is now unconditional (Lee, 17 Dec 2025). For other shapes (squares, chopped-squares, carets), universality is unproven—exceptional vanishing occurs, but computational evidence supports "near-universality" in large Durfee cases (Zhao, 2023, Pak et al., 2013). The interplay between rim-hook combinatorics, character theory, and modular lifting remains active, with ongoing conjectures targeting full universality for certain sequences and new families generated by virtual character techniques (Li, 2017).