Staircase Minimality Theorem

• The Staircase Minimality Theorem defines the k-th staircase partition as the unique minimal 2-regular partition of a triangular number via dominance order.
• It employs combinatorial techniques, including bounds on partition parts and lengths, to prove that only the staircase partition attains minimality.
• The theorem underpins key representation-theoretic results by ensuring every irreducible representation appears in the tensor square, thereby proving Saxl's conjecture.

The Staircase Minimality Theorem characterizes the dominance relations among 2-regular partitions of triangular numbers and constitutes the central combinatorial tool underlying the unconditional proof of Saxl's conjecture. For each positive integer $k$, the $k$-th staircase partition $\rho_k = (k, k-1, \dots, 1)$ defines a fundamental structure within the representation theory of symmetric groups $S_{T_k}$, where $T_k = \frac{k(k+1)}{2}$ is the $k$-th triangular number. The theorem states that among all 2-regular partitions of $T_k$—those with strictly decreasing parts—the staircase $\rho_k$ is the unique element minimal with respect to the dominance order. This minimality property underpins a series of representation-theoretic consequences culminating in the proof that the tensor square $S^{\rho_k}\otimes S^{\rho_k}$ contains every irreducible $S_{T_k}$-representation as a constituent, thereby resolving Saxl's conjecture (Lee, 17 Dec 2025).

1. Formal Statement and Definitions

The $k$-th staircase partition and triangular number are given by

$$\rho_k = (k, k-1, \dots, 1), \qquad T_k = \frac{k(k+1)}{2}.$$

Define the set of 2-regular partitions of $T_k$ as

$$R_{T_k} = \left\{\, \mu = (\mu_1 > \mu_2 > \cdots > \mu_\ell > 0) \,\middle|\, \sum_{i=1}^\ell \mu_i = T_k \, \right\}.$$

The dominance order among partitions $\lambda, \mu \vdash n$ is

$$\lambda \succeq \mu \iff \sum_{i=1}^j \lambda_i \ge \sum_{i=1}^j \mu_i \quad \text{for all } j \ge 1.$$

A partition is dominance-minimal in a set $\mathcal{P}$ if it is dominated by every element of $\mathcal{P}$, uniquely minimal if there is no other element of equal minimality.

Staircase Minimality Theorem:

Among all 2-regular partitions of $T_k$, $\rho_k$ is the unique minimal element in the dominance order. In formal terms,

$\forall\,\mu\in R_{T_k}:\quad \mu \succeq \rho_k$

with equality if and only if $\mu = \rho_k$.

2. Proof Structure: Lemmas and Propositions

The proof involves several combinatorial steps, outlined as follows:

1. Lower bound on parts: Any $\mu=(\mu_1 > \cdots > \mu_\ell) \in R_n$ satisfies $\mu_i \ge \ell - i + 1$ for all $1 \le i \le \ell$, as the minimal configuration is $(\ell, \ell-1, \dots, 1)$.
2. Bound on partition length: For $\mu\in R_{T_k}$, the length satisfies $\ell(\mu) \le k$, since $T_{\ell(\mu)} \le |\mu| = T_k$.
3. Uniqueness at maximal length: If $\mu$ has length $k$, its parts must be exactly $(k, k-1, \dots, 1)$, i.e., $\mu = \rho_k$.
4. Partial sums of the staircase: $$S_j(\rho_k) = \sum_{i=1}^j (k-i+1) = jk - \binom{j}{2}$$ for $1 \le j \le k$.
5. Strict dominance for shorter partitions: If $\ell(\mu) < k$, then for some $j \le \ell(\mu)$,

$S_j(\mu) > S_j(\rho_k)$

with overall $S_j(\mu) \ge S_j(\rho_k)$ for all $j$, meaning $\mu \succ \rho_k$ and not just $\succeq$.

1. Conclusion: Any $\mu\in R_{T_k}$ with length less than $k$ strictly dominates $\rho_k$, and if length equals $k$ then $\mu = \rho_k$.

3. Role in the Proof of Saxl’s Conjecture

The minimality theorem is essential for establishing Kronecker positivity results needed for the proof of Saxl's conjecture.

• Ikenmeyer’s Dominance Positivity:

If $\mu\succeq\rho_k$, then the Kronecker coefficient $g(\mu, \rho_k, \rho_k) \ge 1$. Since all $2$-regular $\mu$ satisfy $\mu\succeq\rho_k$, this positivity applies to every 2-regular partition of $T_k$.

• Modular Saturation:

In characteristic $2$, the projective multiplicity

$$a_\mu = \sum_{\lambda\vdash T_k} g(\lambda, \rho_k, \rho_k) d_{\lambda\mu}$$

satisfies $a_\mu \ge 1$ for all $2$-regular $\mu$ due to $d_{\mu\mu}=1$ (James's theorem) and $g(\mu, \rho_k, \rho_k)\ge 1$. This gives modular saturation for $\rho_k$.

• Lifting to Characteristic Zero:

Applying the Bessenrodt–Bowman–Sutton theorem, modular saturation of the self-conjugate $2$-core $\rho_k$ ensures $g(\lambda, \rho_k, \rho_k)>0$ for every $\lambda\vdash T_k$ in characteristic zero, proving Saxl’s conjecture.

4. Key Definitions and Combinatorial Ingredients

• 2-regular partition: A strictly decreasing sequence of positive integers whose sum is $n$.
• Dominance order: A partial order on partitions based on comparison of partial sums.
• Staircase partition: $\rho_k = (k, k-1, \dots, 1)$, total weight $T_k$.
• Kronecker coefficients: $g(\lambda, \mu, \nu)$ counts multiplicity of the irreducible representation indexed by $\lambda$ in the tensor product of those indexed by $\mu, \nu$.
• Decomposition matrix: In characteristic $p$, $d_{\lambda\mu}$ are decomposition numbers for lifting representations; for 2-regular $\mu$, $d_{\mu\mu}=1$.
• Modular saturation: All relevant projective multiplicities $a_\mu \geq 1$.

5. Explicit Formulas and Theorems

Definition/Theorem	Formula	Application
Dominance order	$$\lambda \succeq \mu \iff \forall j\, \sum_{i=1}^j \lambda_i \geq \sum_{i=1}^j \mu_i$$	Partial sums comparison
Partial sums of $\rho_k$	$S_j(\rho_k)=jk-\binom{j}{2}$	Staircase combinatorics
Projective multiplicity (char. 2)	$$a_\mu = \sum_{\lambda} g(\lambda, \rho_k, \rho_k)d_{\lambda\mu}$$, $d_{\mu\mu}=1$	Modular saturation
Ikenmeyer’s positivity criterion	$\lambda\succeq\rho_k$ or $$\rho_k\succeq\lambda \implies g(\lambda, \rho_k, \rho_k)\geq 1$$	Kronecker positivity
Lifting (Bessenrodt–Bowman–Sutton)	Modular saturation for $2$-core $\gamma$ implies $g(\lambda, \gamma, \gamma) > 0$	Passage to char. $0$

The explicit combinatorics of the minimality theorem provide a transparent route for tracking which partitions appear in the tensor square and for verifying modular and ordinary saturation conditions (Lee, 17 Dec 2025).

6. Uniqueness, Characterization, and Implications

At triangular numbers, the staircase partitions $\rho_k$ are not only dominance-minimal among $2$-regular partitions but are also the sole Kronecker-universal self-conjugate partitions: only for $\rho_k$ does the tensor square contain all irreducible representations of $S_{T_k}$. This provides a complete classification for such universality at these sizes.

A plausible implication is the adoption of the combinatorial minimality approach in related settings where saturation criteria in both modular and ordinary representation theory need to be algorithmically tractable or where explicit constructions of universal constituents are sought.

7. Context and Applicability

The Staircase Minimality Theorem supersedes prior partial results involving dominance and tensor square decompositions. Its deduction chain relies only on explicit combinatorics and three major theorems: Ikenmeyer’s dominance positivity for Kronecker coefficients, James’s result $d_{\mu\mu}=1$ for $2$-regular partitions, and the Bessenrodt–Bowman–Sutton lifting theorem for passage between modular and ordinary representation theory. The strategy generalizes to other settings of projective and tensor product multiplicities indexed by families of minimal partitions or cores (Lee, 17 Dec 2025).