#### “Mathematics Days in Sofia”

Section “Mathematical Logic, Algebra, Number Theory, and Combinatorics”

### Participants

#### Invited Speakers

**Antun Milas**, SUNY Albany, USA**Greta Panova**, University of Southern California, USA**Natasha Rozhkovskaya**, Kansas State University, USA**Plamen Koshlukov**, State University of Campinas, Brazil**Sihem Mesnager**, Universities of Paris VIII, France**Tinko Tinchev**, Sofia University “St. Kliment Ohridski”, Bulgaria

#### Contributors

**Alaa Abouhalaka**, Çukurova University, Turkiye**Assia Rousseva**, Sofia University “St. Kliment Ohridski”, Bulgaria**Azniv Kasparian**, Sofia Universiy “St. Kliment Ohridski”, Bulgaria**Dimiter Dobrev**, Institute of Mathematics and Informatics, Bulgaria**Elitza Hristova**, Institute of Mathematics and Informatics, Bulgaria**Emiliyan Rogachev**, Sofia University “St. Kliment Ohridski”, Bulgaria**Hristo Iliev**, American University in Bulgaria and Institute of Mathematics and Informatics, Bulgaria**Ibrahim Senturk**, Ege University, Turkiye**Ivan Chipchakov**, Institute of Mathematics and Informatics, Bulgaria**Ivanka Nikolova**, University of Pittsburgh, USA**Marin Genov**, Institute of Mathematics and Informatics, Bulgaria**Pancho Beshkov**, Sofia University “St. Kliment Ohridski”, Bulgaria**Şehmus Fındık**, Çukurova University, Turkiye**Tatiana Todorova**, Sofia Universiy “St. Kliment Ohridski”, Bulgaria**Vesselin Drensky**, Institute of Mathematics and Informatics, Bulgaria**Youngook Choi**, Yeungnam University, Republic of Korea

### Abstracts

The aim of this talk is to examine the interplay between jet and arc algebras and vertex algebras. Our focus will be on identifying specific classes of vertex algebras that display the strongest connection to this relationship. To this end, we will consider examples such as principal subspaces, as well as vertex algebras that emerge from 4d/2d CFT correspondences. Throughout our talk, we will discuss combinatorial aspects and asymptotic properties of graded dimensions.

A large part of Algebraic Combinatorics studies discrete structures from Representation Theory with combinatorial methods. The flagship hook-length formula counts the number of Standard Young Tableaux, which also give the dimension of the irreducible Specht modules of the Symmetric group. The beautiful Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, which can formally explain the beauty we see and the difficulties we encounter in finding further formulas and “combinatorial interpretations”. As a proof of concept, we show that the square of an irreducible character of the symmetric group cannot have any positive combinatorial formula as its computation is as hard as the polynomial time hierarchy.

We study the effect of linear transformations on quantum fields, with the main example of application to vertex operator presentations of Hall-Littlewood polynomials.

The construction is illustrated with examples that include certain versions of multiparameter symmetric functions, Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature. Linear transformations of quantum fields effectively describe preservation of commutation relations, stability, explicit combinatorial formulas and generating functions, polynomial tau functions of the KP and the BKP hierarchy.

The description of the group gradings on an algebra is rather important problem in Ring theory. The classification of the group gradings on matrix algebras (Bahturin, Sehgal, Zaicev) plays a prominent role in PI theory. The gradings on upper triangular matrix algebras were proved to be elementary, by Valenti and Zaicev. The elementary gradings on the upper triangular matrices were classified by Di Vincenzo, Koshlukov, Valenti.

Here we study block-triangular matrix algebras: these generalise the full matrix algebras and the upper triangular ones, the latter two lying on the “opposite” ends of block-triangular algebras. Valenti and Zaicev described the group gradings on such algebras provided the group is finite and abelian, and the field is algebraically closed and of characteristic 0. Yukihide recently obtained that every grading on these algebras comes from an elementary grading on a block-triangular algebra and a division grading on a full matrix algebra, provided that the Jacobson radical is homogeneous in the grading. He proved, moreover, that this is the case for arbitrary grading groups if the field is either of characteristic 0 or of characteristic p, larger than the dimension of the algebra.

We prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. This yields the classification of the group gradings on these algebras and confirms a conjecture made by Valenti and Zaicev in 2007.

(Joint results with D. Silva and J. Galdino.)

This talk is devoted to functions over finite fields (Boolean and p-ary functions) as the main components in the general context of protection information theory. Those functions equipped with some cryptographic properties have attracted much attention in the literature, and many activities have been carried out on those functions during approximately five decencies. We shall introduce and discuss these functions by presenting mathematical tools to handle their main cryptographic properties. We shall highlight the related mathematical problems (concerning equations and exponential sums over finite fields) aiming to generate secure families in symmetric cryptography and use them to design linear codes for various applications. We shall focus on specific examples of problems, present a selection of very recent (2020-2022) mathematical achievements, and present some perspectives.