Anatol N. Kirillov
We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the Littlewood--Richardson numbers.
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We prove an asymptotic saturation-type version of Rota's basis conjecture. It relies on the connection of Tao's slice rank with unstable tensors from geometric invariant theory.
A short proof of Tuza's conjecture for weak saturation in hypergraphs
研究超图弱饱和数\(\mathrm{wsat}(n,H)\)，Shapira和Tyomkyn已证图扎猜想，本文给出该猜想更简短证明。
Proof of the Newell-Littlewood saturation conjecture
本文将蜂窝方法推广到莫比乌斯带上，从而证明了 Newell-Littlewood 数的饱和猜想。
Degreewise Cut-Semigroup Saturation for $K_5$-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture
该论文证明了 $K_5$-无 minors 图的齐次割半群在高度 $k$ 的饱和性与平面 $k$-图的 $k$-边可染色性等价，从而将割多面体正规性猜想与 Seymour 平面边染色猜想联系起来。
In this exposition we give a simple and complete treatment of A. Knutson and T. Tao's recent proof (http://front.math.ucdavis.edu/math.RT/9807160) of the saturation conjecture, which asserts that the Littlewood-Richardson semigroup is saturated. The main tool is Knutson and Tao's hive model for Berenstein-Zelevinsky polytopes. In an appendix of W. Fulton it is shown that the hive model is equivalent to the original Littlewood-Richardson rule.
We introduce the honeycomb model of BZ polytopes, which calculate Littlewood-Richardson coefficients, the tensor product rule for GL(n). Our main result is the existence of a particularly well-behaved honeycomb with given boundary conditions (choice of triple of representations to be tensored together). This honeycomb is necessarily integral, which proves the "saturation conjecture", extending results of Klyachko to give a complete answer to which L-R coefficients are positive. This in turn has as a consequence Horn's conjecture from 1962 characterizing the spectrum of the sum of two Hermitian…
We provide counter-examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P-hardness of computing the Kronecker coefficients. Both results rely on the connections between the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups: Murnaghan's reduced Kronecker coefficients. An app…
A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. Determining the saturation number of complete partite graphs is one of the most important problems in the study of saturation number. The value of $sat(n,K_{2,2})$ was shown to be $\lfloor\frac{3n-5}{2}\rfloor$ by Ollmann, and a shorter proof was later given by Tuza. For $K_{2,3}$, there has been a series …
Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = Θ(n)$. They found two 0-1 matrices $P$ for …
Asymmetric induced saturation
研究对于哪些图H存在至少有一条边且无H诱导子图的图G，使得删除G的任意边都会产生H的诱导副本，并证明该存在性对多种图类成立。
Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $\mathcal{C}$. Similarly to classical saturation functions, define $\mathrm{sat}_t(n, \mathcal{C})$ to be the minimum number of edges in a $(\mathcal{C},t)$ saturated graph. Let $\mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. In this paper we consider a variety of colored saturation proble…
Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.
Projective and external saturation problem for posets
本文提出偏序集饱和问题的射影与外饱和两个变体，研究其饱和数性质，并猜想所有有限偏序集的外饱和数有界，且在特殊情形下给出证明。
Poset saturation of unions of chains
本文研究由两两不可比的链构成的偏序集的导出饱和数，提出关于其线性与常数阶上界的两个猜想，并在等长链及 $2C_k+C_1$ 等特殊情形下予以证明，同时给出无穷多例验证常数阶界无需链长各异。
The induced saturation problem for posets
本文研究偏序集的诱导饱和问题，将已知的二分下界从对数级改进为 $\min\{2\sqrt{n}, n/2+1\}$，并利用有向图 Turán 型结果证明，同时验证了关于一类偏序集的线性下界猜想。
The saturation theorem of [Knutson-Tao '99] concerns the nonvanishing of Littlewood-Richardson coefficients. In combination with work of [Klyachko '98], it implies [Horn '62]'s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization [Friedland '00] to majorized sums of Hermitian matrices. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmanni…
We report about results revolving around Kostka-Foulkes and parabolic Kostka polynomials and their connections with Representation Theory and Combinatorics. It appears that the set of all parabolic Kostka polynomials forms a semigroup, which we call {\it Liskova semigroup}. We show that polynomials frequently appearing in Representation Theory and Combinatorics belong to the Liskova semigroup. Among such polynomials we study rectangular $q$-Catalan numbers; generalized exponents polynomials; principal specializations of the internal product of Schur functions; generalized $q$-Gaussian polynomi…
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