Depth-Kronecker Parameterization

• Depth-Kronecker Parameterization is a combinatorial and algebraic framework that expresses Kronecker square coefficients as universal polynomials based on connected skew-diagrams.
• It leverages explicit formulas constructed from special rim-hook tabloids and polynomial subdiagram counts, refining classical symmetric function methods.
• The framework underpins stability phenomena and asymptotic behavior in Kronecker coefficients, enhancing computation and classification in representation theory.

Depth-Kronecker Parameterization refers to a combinatorial and algebraic framework for expressing Kronecker coefficients—specifically, the multiplicity of irreducible characters within Kronecker squares of symmetric group representations—in terms of explicit, finite families of combinatorial objects and explicit polynomials. This parameterization provides a closed formula for the Kronecker square coefficients in terms of universal polynomials depending on the depth of the partition and the occurrence statistics of certain connected skew diagrams, facilitating detailed combinatorial analysis, asymptotic formulae, and stability results for Kronecker coefficients (Vallejo, 2013).

1. Definitions and Foundational Concepts

Given partitions $\lambda, \mu \vdash n$ and $\nu$ of size $d\le n$, the Kronecker coefficient $g(\chi^{\lambda}, \chi^{\mu}, \chi^{\nu})$ denotes the multiplicity of the irreducible character $\chi^{\nu}$ in the decomposition of the tensor product $\chi^{\lambda} \otimes \chi^{\mu}$ as a character of $S_n$. For the context of this parameterization, attention focuses on the case where $\lambda = \mu$ (the Kronecker square), and sometimes further specializes to staircase partitions.

The depth of a partition $\nu = (\nu_1, \nu_2, ..., \nu_r) \vdash d$ is defined as $\mathrm{depth}(\nu) = d - \nu_1$. Connected skew-diagrams $\gamma = \lambda/\alpha$ of size at most the depth factor crucially into the parameterization scheme: for each such isomorphism class $D$, let $r_\lambda(D)$ be the number of $\lambda$-removable subdiagrams isomorphic to $D$.

2. The Depth-Kronecker Parameterization Theorem

The main result establishes the existence of a universal polynomial $P_\nu$ for each $\nu \vdash d$, such that for $\lambda \vdash n$ with $n > d + \nu_1$,

$$g(\chi^{\lambda}, \chi^{\lambda}, \chi^{(n-d, \nu)}) = P_\nu\big(r_\lambda(D_1), r_\lambda(D_2), ..., r_\lambda(D_m)\big)$$

where $D_i$ runs over the isomorphism classes of connected skew-diagrams of sizes up to $d$. Each $r_\lambda(D_i)$ is itself a polynomial (with rational coefficients) in subdiagram-counting functions of $\lambda$.

This parameterization stems from a diagrammatic refinement of the Robinson–Taulbee method, employing the Jacobi–Trudi identity, Littlewood–Richardson rule, and combinatorics of special rim-hook tabloids, augmented by systematic encoding of removals via connected subdiagrams (Vallejo, 2013).

3. Explicit Construction of the Parameterizing Polynomial

The polynomial $P_\nu$, for each $\nu$, is constructed as follows:

• For each $k \leq d$ and each special rim-hook tabloid $T$ of shape $\nu$ with excess $e(T) = k$, assign a sign $\mathrm{sgn}(T)$.
• For each diagram class $D$ of size $k$, let $\mathrm{Ir}(D, D; \tau(T))$ count certain pairs of Littlewood–Richardson multitableaux with prescribed content.
• Summing over all $k, T, D$ and incorporating the universal polynomials $P_D$ (in the $r_\lambda(D_i)$), one obtains

$$P_\nu = \sum_{k=0}^{d} \sum_{T \in \mathrm{SBST}(\nu),\, e(T)=k} \mathrm{sgn}(T) \sum_{D:\,|D|=k} \mathrm{Ir}(D, D; \tau(T))\,P_D(x_{D_1},...,x_{D_m}),$$

where SBST denotes the set of special rim-hook tabloids.

• Each $P_{D}$ is a universal polynomial expressing $r_\lambda(D)$ in terms of numbers of connected, $\lambda$-removable subdiagrams of smaller or equal size.

Upon substitution of combinatorial data for a specific $\lambda$, evaluation yields the Kronecker coefficient (Vallejo, 2013).

4. Piecewise-Polynomiality and Asymptotic Features

For particularly structured sequences of partitions, such as staircases $P_k = (k, k-1, ..., 1)$ of size $n_k = k(k+1)/2$, the composition

$$k \mapsto g(\chi^{P_k}, \chi^{P_k}, \chi^{(n_k - d, \nu)})$$

is itself a rational piecewise-polynomial function in $k$. For sufficiently large $k \geq t(\nu)$ (an explicit function of $\nu$), this is an ordinary polynomial of degree $d$ with leading coefficient equal to the number of standard tableaux of $\nu$.

For each connected class $D$ of size $e \le d$, the statistic $f_D(k) := r_{P_k}(D)$ is a piecewise-polynomial function, vanishing for small $k$, and eventually agreeing with a polynomial in $k$ of degree up to $e$. Explicit tables of these polynomials for small $d$ verify exact agreement with direct Kronecker coefficient computations (Vallejo, 2013).

5. Stability Phenomena

The Depth-Kronecker Parameterization also establishes new stability properties for Kronecker coefficients. Because the coefficients depend only on the finite set of statistics $r_\lambda(D)$, any transformation of $\lambda$ and/or $\mu$ that leaves these unchanged preserves the Kronecker coefficient. For example, if $\lambda, \mu$ satisfy

$$\lambda_i - \lambda_{i+1} \geq d,\quad \mu_i - \mu_{i+1} \geq d,$$

for $i\le \min\{\mathrm{depth}(\nu),\,\ell(\mu)\}$, then

$$g(\chi^\lambda, \chi^\mu, \chi^\nu) = g(\chi^{\lambda+(k,k)}, \chi^{\mu+(k,k)}, \chi^\nu)$$

for all $k\ge 0$, where $\lambda + (k,k)$ means adding $k$ to rows $i$ and $i+1$ of $\lambda$. This generalizes Murnaghan's classical stability by revealing wide classes of invariant directions in the space of partitions (Vallejo, 2013).

6. Applications and Limitations

Within algebraic combinatorics and the study of Kronecker coefficients, the Depth-Kronecker Parameterization enables systematic computation, combinatorial classification, and explicit tabulation of Kronecker squares. It exposes the polynomial structure and stability underlying these multiplicities, supporting further advances in understanding their asymptotic and extremal behaviors. However, the framework is specific to partitions of bounded depth and does not necessarily provide effective closed forms for all pairs or triples of partitions, especially those of unbounded depth or irregular structure (Vallejo, 2013).

7. Context within Representation Theory and Related Work

The Depth-Kronecker Parameterization concretizes longstanding insights that Kronecker coefficients for certain families of partitions are governed by finite combinatorial data. The explicit closed-form results for staircase partitions and bounded-depth cases give substantial evidence for generalized stability conjectures (e.g., Saxl's conjecture). The parameterization arises from augmentations of the Robinson–Taulbee method and is closely related to developments in diagrammatic approaches, symmetric functions, and tableau combinatorics. It provides a canonical bridge from combinatorial statistics to the explicit calculation of structural multiplicities in $S_n$ representation theory (Vallejo, 2013).