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C. A. Pallikaros

Publications and source records attributed to C. A. Pallikaros.

8 recordsLinked to original sources
arXiv2024

Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field

We determine the complete degeneration picture inside the variety of nilpotent associative algebras of dimension 3 over an algebraically closed field of characteristic not equal to 2. Comparing with the discussion in [Ivanova N.M. and Pallikaros C.A., Degenerations of complex associative algebras of dimension three via Lie and Jordan algebras, {\it Advances in Group Theory and Applications}, 18 (2024), 41-79], for some of the arguments in the present article we needed to develop alternative techniques which are now valid over an arbitrary algebraically closed field. There is a dichotomy of cases concerning the results obtained, corresponding to whether the characteristic of the field is 2 or not.

math.RA↗
arXiv2022

Degenerations of complex associative algebras of dimension three via Lie and Jordan algebras

Let $\boldsymbolΛ_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbolΛ_3(\mathbb C)$ `by change of basis'. We determine the complete degeneration picture inside the algebraic subset $\mathcal A^s_3$ of $\boldsymbolΛ_3(\mathbb C)$ consisting of associative algebra structures via the corresponding information on the algebraic subsets $\mathcal L_3$ and $\mathcal J_3$ of $\boldsymbolΛ_3(\mathbb C)$ of Lie and Jordan algebra structures respectively. This is achieved with the help of certain $G$-module endomorphisms $ϕ_1$, $ϕ_2$ of $\boldsymbolΛ_3(\mathbb C)$ which map $\mathcal A^s_3$ onto algebraic subsets of $\mathcal L_3$ and $\mathcal J_3$ respectively.

math.RA↗
arXiv2021

Minimal determining sets for certain $W$-graph ideals

We consider Kazhdan-Lusztig cells of the symmetric group $S_n$ containing the longest element of a standard parabolic subgroup of $S_n$. Extending some of the ideas in [Beitr{ä}ge zur Algebra und Geometrie, 59 (2018), no. 3, 523-547] and [Journal of Algebra and Its Applications, 20 (2021), no. 10, 2150181], we determine the rim of some additional families of cells and also of certain induced unions of cells. These rims provide minimal determining sets for certain $W$-graph ideals introduced in [Journal of Algebra, 361 (2012), 188-212].

math.RT↗
arXiv2019

On ordered $k$-paths and rims for certain families of Kazhdan-Lusztig cells of $S_n$

For a composition $λ$ of $n$ we consider the Kazhdan-Lusztig cell in the symmetric group $S_n$ containing the longest element of the standard parabolic subgroup of $S_n$ associated to $λ$. In this paper we extend some of the ideas and results in [{Beitr{ä}ge} zur Algebra und Geometrie, \textbf{59} (2018), no.~3, 523--547]. In particular, by introducing the notion of an ordered $k$-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.

math.RT↗
arXiv2018

On embedding certain Kazhdan--Lusztig cells of $S_n$ into cells of $S_{n+1}$

In this paper, we consider a particular class of Kazhdan-Lusztig cells in the symmetric group $S_n$, the cells containing involutions associated with compositions $λ$ of $n$. For certain families of compositions we are able to give an explicit description of the corresponding cells by obtaining reduced forms for all their elements. This is achieved by first finding a particular class of diagrams $\mathcal{E}^{(λ)}$ which lead to a subset of the cell from which the remaining elements of the cell are easily obtained. Moreover, we show that for certain cases of related compositions $λ$ and $\hatλ$ of $n$ and $n+1$ respectively, the members of $\mathcal{E}^{(λ)}$ and $\mathcal{E}^{(\hatλ)}$ are also related in an analogous way. This allows us to associate certain cells in $S_n$ with cells in $S_{n+1}$ in a well-defined way, which is connected to the induction and restriction of cells.

math.RT↗
arXiv2017

Describing certain Lie algebra orbits via polynomial equations

Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of structure constants corresponding to $\mathfrak{h}_3$ and $\mathfrak g$ respectively as an algebraic set giving in each case a set of polynomials for which the orbit closure is the set of common zeros. Working over an arbitrary infinite field, this description enables us to give an alternative way, using the definition of an irreducible algebraic set, of obtaining all degenerations of $\mathfrak{h}_3$ and $\mathfrak g$ (the degeneration from $\mathfrak g$ to $\mathfrak{h}_3$ being one of them).

math-ph↗