Ernesto Vallejo
In this note we make explicit a stability property for Kronecker coefficients that is implicit in a theorem of Y. Dvir. Even in the simplest nontrivial case this property was overlooked despite of the work of several authors. As an application we give a new formula for some Kronecker coefficients.
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We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker coefficients using stratifications of the cell modules of the partition algebra.
Some unimodal sequences of Kronecker coefficients
本文猜想并证明广义 Kronecker 系数序列的单峰性（对至多两列划分），利用硬 Lefschetz 性质，并探讨更一般的 Lefschetz 性质及对 Alon–Tarsi 猜想高维类比的启示。
The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture
本文研究了确定Kronecker系数正性的问题，通过半群性质和广义块方法探讨Saxl猜想中不可约表示的分解问题。
Highlighting the use of critical classes, we consider constituents in Kronecker products, in particular of spin characters of the double covers of the symmetric and alternating groups. We apply results from the spin case to find constituents in Kronecker products of characters of the symmetric groups. Via this tool, we make progress on the Saxl conjecture; this claims that for a triangular number $n$, the square of the irreducible character of the symmetric group $S_n$ labelled by the staircase contains all irreducible characters of $S_n$ as constituents. We find a large number of constituents…
Using a recursion formula due to Dvir, we obtain information on maximal and almost maximal components in Kronecker products of characters of the symmetric groups. This is applied to confirm a conjecture made by Bessenrodt and Kleshchev in 1999, which classifies all such Kronecker products with only three or four components.
We provide a classification of multiplicity-free inner tensor products of irreducible characters of symmetric groups, thus confirming a conjecture of Bessenrodt. Concurrently, we classify all multiplicity-free inner tensor products of skew characters of the symmetric groups. We also provide formulae for calculating the decomposition of these tensor products.
We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups S_n contain all irreducibles as their constituents. Our main result is that they contain representations corresponding to hooks and two row Young diagrams. For that, we develop a new sufficient condition for the positivity of Kronecker coefficients in terms of characters, and use combinatorics of rim hook tableaux combined with known results on unimodality of certain partition functions. We also present connections and speculations on random characters of S_n.
We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(α,β,γ)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(Nα,Nβ,Nγ)>0$ implies $\overline{g}(α,β,γ)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(α,β,γ)$, over all $|α|+|β|+|γ| = n$. Finally, we show that computing $\overline{g}(λ,μ,ν)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(λ,μ,ν)$ is in unary.
The Newton polytope of the Kronecker product
本文研究了两个 Schur 函数的 Kronecker 积，利用 Horn 不等式证明了其单项式展开具有饱和 Newton 多面体的猜想特例，并导出了 Kronecker 系数正性的必要条件。
Field-independent Kronecker-plethysm isomorphisms
本文构造了域无关的SL2-等变同构，推广了Hermite互反律等经典同构，并证明了Kronecker系数与plethysm系数的一致性，从而在任意特征下证明了q-二项式系数的单峰性。
We show that the Kronecker coefficients (the Clebsch-Gordan coefficients of the symmetric group) indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients in…
We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound via the characters of $S_n$. We apply these and other results to generalize Sylvester's unimodality of the $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in $q$: we derive explicit sharp bounds on the differences of their consecutive coefficients.
The Kronecker product of two Schur functions $s_μ$ and $s_ν$, denoted by $s_μ*s_ν$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions $μ$ and $ν$. The coefficient of $s_λ$ in this product is denoted by $γ^λ_{μν}$, and corresponds to the multiplicity of the irreducible character $χ^λ$ in $χ^μχ^ν.$ We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for $s_λ[XY]$ to find closed formulas for the Kronecker coefficients $γ^λ_{μν}$ when $λ$ is an…
We present a lower bound on the Kronecker coefficients for tensor squares of the symmetric group via the characters of~$S_n$, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in~$q$ to derive sharp bounds on the differences of their consecutive coefficients. We then derive effective asymptotic lower bounds for a wider class of Kronecker coefficients.
In this paper we apply a method of Robinson and Taulbee for computing Kronecker coefficients together with other ingredients and show that the multiplicity of each component in a Kronecker square can be obtained from an evaluation of a certain polynomial, which depends only on the component and is computed combinatorially. This polynomial has as many variables as the set of isomorphism classes of connected skew diagrams of size at most the depth of the component. We present two applications. The first is a contribution to Saxl conjecture, which asserts that the Kronecker square of the staircas…
We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n θ$, in the case where $θ$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.
We study the complexity of computing Kronecker coefficients $g(λ,μ,ν)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction …
Cancellation and regularity for planar, 3-connected Kronecker products
本文研究平面、3-连通图的 Kronecker 积，证明了该类积满足消去律，并分类了其中的面正则与顶点正则图，同时刻画了可双重表示为 Cartesian 积的平面图。
In this paper we give a combinatorial interpretation for the coefficient of $s_ν$ in the Kronecker product $s_{(n-p,p)}\ast s_λ$, where $λ=(λ_1, ..., λ_{\ell(λ)})\vdash n$, if $\ell(λ)\geq 2p-1$ or $λ_1\geq 2p-1$; that is, if $λ$ is not a partition inside the $2(p-1)\times 2(p-1)$ square. For $λ$ inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all $λ$ when $n>(2p-2)^2$. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker produ…
Algebraic connectivity of Kronecker products of line graphs
该研究刻画了线图与完全图的克罗内克积代数连通度为m-1的树的特征，给出其为拉普拉斯整图的充要条件，并讨论了两类树对应乘积图的代数连通度。
Quantum complexity of the Kronecker coefficients
本文证明Kronecker系数可表示为QMA验证器接受见证空间的维数，从而将其近似与正性判定纳入量子复杂性框架，并给出近似归一化Kronecker系数的高效量子算法。
Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
本文在固定边数二部带状图张成的代数 Hilbert 空间上构造可解量子系统，以对称群特征标和 Kronecker 系数刻画谱与简并，并给出 Kronecker 系数的组合格维数解释。
Eigenvalues of Hypergraph Products and Reciprocal Eigenvalue Property
本文研究了超图的连接、克罗内克和冠状乘积的邻接、拉普拉斯和无向拉普拉斯特征值，探讨了这些谱在操作下的行为，并扩展了互为特征值性质到超图，证明幂超树不满足该性质。
Classification and Construction of Planar, 3-Connected Kronecker Products
该研究完成了平面3-连通Kronecker积图的完整分类，证明其均为$H\wedge K_2$形式，给出构造性证明与平面图$H$的精细刻画，并补充了相关积图的3-多面体性讨论。
In a recent paper, the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products that have as consequences several versions of the Pieri rule for this new basis of symmetric functions. In addition, we give several applications of the products studied in this paper.
Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree distribution) with efficiently calculable ground-truth solutions to the desired output. Reproducibility for current generators used in benchmarking are somewhat lacking in this respect due to their randomness: the output of a desired graph analytic can only be compared to expected values and not exact ground truth. Nonstochastic Kronecker product graphs meet these des…
We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.
A Unified Approach to Unimodality of Gaussian Polynomials
提出基于计算机代数的新方法，通过闭式系数和柱形代数分解精确判定$q$-二项式系数的严格单峰性，并证明Stanley和Zanello猜想的两个额外情形。
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