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Mitja Mastnak

Publications and source records attributed to Mitja Mastnak.

At least 19 recordsLinked to original sources
arXiv2026

Fence Posets, Good Gradings and Frobenius Maximal Parabolics

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of $\mathfrak{gl}_n$ in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element $e$ lies in $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

math.RT↗
arXiv2026

The Center of the Temperley-Lieb Algebra

We compute the dimension of the center of the Temperley--Lieb algebra $\operatorname{TL_n}(\delta)$ over a field of characteristic zero for every nonzero value of the parameter $\delta$. The proof uses the cellular filtration by cup number, together with known facts about the representation theory of the Temperley--Lieb algebra, especially the structure of its standard modules and their radicals. Dilation and compression maps compare the induced graded pieces of the center at levels $n$ and $n-2$, giving an upper bound of one for each such piece. A deformation argument gives the matching lower bound, and hence $ \dim Z(\operatorname{TL}_n(\delta))=1+\Bigl\lfloor \frac{n}{2}\Bigr\rfloor$. We also prove that every central element is fixed by the canonical anti-automorphism and by the natural diagram-reflection automorphism. Finally, we give a congruence criterion for the trivial-radical case and record a Gram-matrix computation for leading terms.

math.QA↗
arXiv2024

Groups of matrices with approximately submultiplicative spectra

We say that a semigroup of matrices has a submultiplicative spectrum if the spectrum of the product of any two elements of the semigroup is contained in the product of the two spectra in question (as sets). In this note we explore an approximate version of this condition.

math.RT↗
arXiv2024

Invariant embeddings and ergodic obstructions

We consider the following question: Let $\mathcal{A}$ be an abelian self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{H}$. Assume that $\mathcal{A}$ is invariant under conjugation by a unitary operator $U$, i.e., $U^* AU$ is in $\mathcal{A}$ for every member $A$ of $\mathcal{A}$. Is there a maximal abelian self-adjoint algebra containing $\mathcal{A}$, which is still invariant under conjugation by $U$? The answer, which is easily seen to be yes in finite dimensions, is not trivial in general. We prove affirmative answers in special cases including the one where $\mathcal{A}$ is generated by a compact operator. We also construct a counterexample in the general case, whose existence is perhaps surprising.

math.FA↗
arXiv2024

On Hopf algebras whose coradical is a cocentral abelian cleft extension

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_ξ$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_ξ \simeq B_{ξ^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_ξ $ as algebra for $ξ\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216.

math.QA↗
arXiv2024

Invariant embeddings and weighted permutations

We prove that for any fixed unitary matrix $U$, any abelian self-adjoint algebra of matrices that is invariant under conjugation by $U$ can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by $U$. We use this result to analyse the structure of matrices $A$ for which $A^*A$ commutes with $AA^*$, and to characterize matrices that are unitarily equivalent to weighted permutations.

math.RA↗
arXiv2023

On approximate and actual reducibility of matrix groups

We introduce the notions of $\varepsilon$-approximate fixed point and weak $\varepsilon$-approximate fixed point. We show that for a group of unitary matrices even the existence of a nontrivial weak $\varepsilon$-approximate fixed point for sufficiently small $\varepsilon$ gives an actual nontrivial common eigenvector. We give estimates for $\varepsilon$ in terms of the size $n$ of matrices and prove that the dependence is polynomial. Moreover, we show that the common eigenvector is polynomially close to the starting weak approximate fixed point.

math.GR↗
arXiv2022

Approximate NFA Universality and Related Problems Motivated by Information Theory

In coding and information theory, it is desirable to construct maximal codes that can be either variable length codes or error control codes of fixed length. However deciding code maximality boils down to deciding whether a given NFA is universal, and this is a hard problem (including the case of whether the NFA accepts all words of a fixed length). On the other hand, it is acceptable to know whether a code is `approximately' maximal, which then boils down to whether a given NFA is `approximately' universal. Here we introduce the notion of a $(1-ε)$-universal automaton and present polynomial randomized approximation algorithms to test NFA universality and related hard automata problems, for certain natural probability distributions on the set of words. We also conclude that the randomization aspect is necessary, as approximate universality remains hard for any fixed polynomially computable $ε$.

cs.FL↗
arXiv2019

A construction which relates c-freeness to infinitesimal freeness

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space $V$ given together with a linear map $Δ: V \to V \otimes V$. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on $V$, and our main result concerns a certain transformation $Δ^{*}$ on such families of multilinear functionals.

math.OA↗
arXiv2019

Partitioning a Symmetric Rational Relation into Two Asymmetric Rational Relations

We consider the problem of partitioning effectively a given symmetric (and irreflexive) rational relation R into two asymmetric rational relations. This problem is motivated by a recent method of embedding an R-independent language into one that is maximal R-independent, where the method requires to use an asymmetric partition of R. We solve the problem when R is realized by a zero-avoiding transducer (with some bound k): if the absolute value of the input-output length discrepancy of a computation exceeds k then the length discrepancy of the computation cannot become zero. This class of relations properly contains all recognizable, all left synchronous, and all right synchronous relations. We leave the asymmetric partition problem open when R is not realized by a zero-avoiding transducer. We also show examples of total wordorderings for which there is a relation R that cannot be partitioned into two asymmetric rational relations such that one of them is decreasing with respect to the given word-ordering.

cs.FL↗
arXiv2018

On finite-dimensional copointed Hopf algebras over dihedral groups

We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group D_m, with m=4a> 11. We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups.

math.QA↗
arXiv2018

Bialgebra Coverings and Transfer of Structure

We introduce the bicategory of bialgebras with coverings (which can be thought of as coalgebra-indexed families of morphisms), and provide a motivating application to the transfer of formulas for primitives and antipode. Additionally, we study properties of this bicategory and various sub-bicategories, and describe some universal constructions. Finally, we generalize Nichols' result on bialgebra quotients of Hopf algebra, which gives conditions on when the resulting bialgebra quotient is a Hopf algebra.

math.RA↗
arXiv2016

Matrix semigroups whose ring commutators have real spectra are realizable

We study matrix semigroups in which ring commutators have real spectra. We prove that irreducible semigroups with this property are simultaneously similar to semigroups of real-entried matrices. We also obtain a structure theorem for compact groups satisfying the property under investigation.

math.RT↗
arXiv2016

Double-ended queues and joint moments of left-right canonical operators on full Fock space

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (A, phi), a family of "(l,r)-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (l,r)-cumulants of a (2d)-tuple (A_1, ..., A_d, B_1, ..., B_d), with A_1, ... , A_d canonical operators on the left and B_1, ... , B_d canonical operators on the right. This extends a known one-sided formula for free cumulants of A_1, ..., A_d, which establishes a basic operator model for the R-transform of free probability.

math.OA↗
arXiv2015

Embedding rationally independent languages into maximal ones

We consider the embedding problem in coding theory: given an independence (a code-related property) and an independent language $L$, find a maximal independent language containing $L$. We consider the case where the code-related property is defined via a rational binary relation that is decreasing with respect to any fixed total order on the set of words. Our method works by iterating a max-min operator that has been used before for the embedding problem for properties defined by length-increasing-and-transitive binary relations. By going to order-decreasing rational relations, represented by input-decreasing transducers, we are able to include many known properties from both the noiseless and noisy domains of coding theory, as well as any combination of such properties. Moreover, in many cases the desired maximal embedding is effectively computable.

cs.FL↗
arXiv2015

On rigidity of Nichols algebras

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket.

math.RA↗
arXiv2015

Isometries of the Toeplitz Matrix Algebra

We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular, that every such an isometry is a complete isometry and that a unital linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is necessarily an algebra homomorphism.

math.FA↗