arXiv subjects

D. V. Bulgakova

Publications and source records attributed to D. V. Bulgakova.

5 recordsLinked to original sources

arXiv2022

Construction of the traceless projection of tensors via the Brauer algebra

We describe how traceless projection of tensors of a given rank can be constructed in a closed form. On the way to this goal we invoke the representation theory of the Brauer algebra and the related Schur-Weyl dualities. The resulting traceless projector is constructed from purely combinatorial data involving Young diagrams. By construction, the projector manifestly commutes with the symmetric group and is well-adapted to restrictions to $GL$-irreducible tensor representations. We develop auxiliary computational techniques which serve to take advantage of the obtained results for applications. The proposed method of constructing traceless projectors leads to a particular central idempotent in the semisimple regime of the Brauer algebra.

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arXiv2021

On balanced and abelian properties of circular words over a ternary alphabet

We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering the minimal bound $3$ for abelian complexity of balanced circular words over a ternary alphabet, we provide a classification of all circular words over a ternary alphabet with abelian complexity subject to this bound. This result also allows us to construct an uncountable set of bi-infinite aperiodic words with abelian complexity equal to $3$.

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arXiv2019

Fusion procedure for the walled Brauer algebra

We establish two versions of the fusion procedure for the walled Brauer algebras. In each of them, a complete system of primitive pairwise orthogonal idempotents for the walled Hecke algebra is constructed by consecutive evaluations of a rational function in several variables on contents of standard walled tableaux.

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arXiv2019

On the structure of modules over walled Brauer algebra via normal form and random walks

We analyze cyclic cell modules over walled Brauer algebra in terms of a certain normal form. The latter allows us to decompose the algebra into the generating set and annihilator ideal of a certain cyclic vector. In addition, we show that the numbers of reduced basis monomials of given length coincide with those for the symmetric group. For the semisimple case we utilize the theory of differential posets to calculate the dimensions of modules in terms of the paths in Bratelli diagram. It turns out that the number of primitive idempotents is the same as for the symmetric group.

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arXiv2017

Bimodule structure of the mixed tensor product over $U_{q} s\ell(2|1)$ and quantum walled Brauer algebra

We study a mixed tensor product $\mathbf{3}^{\otimes m} \otimes \mathbf{\overline{3}}^{\otimes n}$ of the three-dimensional fundamental representations of the Hopf algebra $U_{q} s\ell(2|1)$, whenever $q$ is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective $U_{q} s\ell(2|1)$-module with the generating modules $\mathbf{3}$ and $\mathbf{\overline{3}}$ are obtained. The centralizer of $U_{q} s\ell(2|1)$ on the chain is calculated. It is shown to be the quotient $\mathscr{X}_{m,n}$ of the quantum walled Brauer algebra. The structure of projective modules over $\mathscr{X}_{m,n}$ is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over $\mathscr{X}_{m,n}$. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over $\mathscr{X}_{m,n}\boxtimes U_{q} s\ell(2|1)$. We give an explicit bimodule structure for all $m,n$.

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