Related papers: Properties of four partial orders on standard Youn…
We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…
For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[\delta]$. We construct modules $\{ \Delta_{\underline{\mu}} \}_{\underline{\mu}}$ for $P^l_n$ over $\mathbb{Z}[\delta]$, and hence over any integral domain…
The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be…
We find a group-theoretical condition under which a twist of a group algebra, in Movshev's way, admits an integral Hopf order. Let $K$ be a (large enough) number field with ring of integers $R$. Let $G$ be a finite group and $M$ an abelian…
Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups -- coordinate functions, differential…
The goal of this thesis is to prove that $\pi_4(S^3) \simeq \mathbb{Z}/2\mathbb{Z}$ in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory,…
We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_\nu[h_\lambda]$ and $s_{\nu}[e_{\lambda}]$, for any partitions $\lambda$ and…
The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen--Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen--Macaulay poset of…
We classify finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradcial is isomorphic to the smallest non-pointed basic Hopf algebra, under the assumption that the diagrams are strictly…
Let U be a unipotent group over the field of complex numbers C, acting on a complex algebraic variety X. Assume that there exists a surjective morphism of complex algebraic varieties f: X --> Y whose fibres are orbits of U. We show that if…
The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of…
We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we…
For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under…
We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are…
We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.
A homotopy commutative algebra, or $C_{\infty}$-algebra, is defined via the Tornike Kadeishvili homotopy transfer theorem on the vector space generated by the set of Young tableaux with self-conjugated Young diagrams. We prove that this…
We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of…
We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we…
We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with…
We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra $S$ (the Landweber-Novikov…