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Justyna Kosakowska

Publications and source records attributed to Justyna Kosakowska.

17 recordsLinked to original sources
arXiv2025

Abelian p-groups with a fixed elementary subgroup or with a fixed elementary quotient

In his 1934 paper, G.\ Birkhoff poses the problem of classifying pairs $(G,U)$ where $G$ is an abelian group and $U\subset G$ a subgroup, up to automorphisms of $G$. In general, Birkhoff's Problem is not considered feasible. In this note, we fix a prime number $p$ and assume that $G$ is a direct sum of cyclic $p$-groups and $U\subset G$ is a subgroup. Under the assumption that the factor group $G/U$ is an elementary abelian $p$-group, we show that the pair $(G,U)$ always has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),\mathbb Z/(p^n))$ or $(\mathbb Z/(p^n), (p))$. Surprisingly, in the dual situation we need an additional condition. If we assume that $U$ itself is an elementary subgroup of $G$, then we show that the pair $(G,U)$ has a direct sum decomposition into pairs of type $(\mathbb Z/(p^n),0)$ or $(\mathbb Z/(p^n), (p^{n-1}))$ if and only if $G/U$ is a~direct sum of cyclic $p$-groups. We generalize the above results to modules over commutative discrete valuation rings.

math.GR↗
arXiv2020

The socle tableau as a dual version of the Littlewood-Richardson tableau

Like the LR-tableau, a socle tableau is given as a skew diagram with certain entries. Unlike in the LR-tableau, the entries in the socle tableau are weakly increasing in each row, strictly increasing in each column and satisfy a modified lattice permutation property. In the study of embeddings of a subgroup in a finite abelian $p$-group, socle tableaux occur as isomorphism invariants, they are given by the socle series of the subgroup. We show that each socle tableau can be realized by some embedding. Moreover, the socle tableau of an embedding and the LR-tableau of the dual embedding determine each other.

math.RT↗
arXiv2020

Combinatorial algorithms for binary operations on LR-tableaux with entries equal to 1 with applications to nilpotent linear operators

In the paper we investigate an algorithmic associative binary operation $*$ on the set $\mathcal{LR}_1$ of Littlewood-Richardson tableaux with entries equal to one. We extend $*$ to an algorithmic nonassociative binary operation on the set $\mathcal{LR}_1\times \mathbb{N}$ and show that it is equivalent to the operation of taking the generic extensions of objects in the category of homomorphisms from semisimple nilpotent linear operators to nilpotent linear operators. Thus we get a combinatorial algorithm computing generic extensions in this category.

math.RT↗
arXiv2019

Two Partial Orders for Standard Young Tableaux

In this manuscript we show that two partial orders defined on the set of standard Young tableaux of shape $α$ are equivalent. In fact, we give two proofs for the equivalence of the box order and the dominance order for {tableaux}. Both are algorithmic. The first of these proofs emphasizes links to the Bruhat order for the symmetric group and the second provides a more straightforward construction of the cover relations. This work is motivated by the known result that the equivalence of the two combinatorial orders leads to a description of the geometry of the representation space of invariant subspaces of nilpotent linear operators.

math.RT↗
arXiv2019

Finite direct sums of cyclic embeddings

In this paper we generalize Kaplansky's combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group given in his 1951 book ``Infinite Abelian Groups''. For this we introduce partial maps on Littlewood-Richardson tableaux and show that they characterize the isomorphism types of finite direct sums of such cyclic embeddings.

math.RT↗
arXiv2019

Box moves on Littlewood-Richardson tableaux and an application to invariant subspace varieties

In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.

math.RT↗
arXiv2016

The boundary of the irreducible components for invariant subspace varieties

Given partitions $α$, $β$, $γ$, the short exact sequences $0\to N_α\to N_β\to N_γ\to 0$ of nilpotent linear operators of Jordan types $α$, $β$, $γ$, respectively, define a constructible subset $\mathbb V_{α,γ}^β$ of an affine variety. Geometrically, the varieties $\mathbb V_{α,γ}^β$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $Γ$ of shape $(α,β,γ)$ contributes one irreducible component $\overline{\mathbb V}_Γ$. We consider the partial order $Γ\leq_{\sf bound}^*\widetildeΓ$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetildeΓ}\cap \overline{\mathbb V}_Γ\neq \emptyset$. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of $α$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $β\setminusγ$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.

math.RT↗
arXiv2016

Operations on Arc Diagrams and Degenerations for Invariant Subspaces of Linear Operators. Part II

For a partition $β$, denote by $N_β$ the nilpotent linear operator of Jordan type $β$. Given partitions $β$, $γ$, we investigate the representation space ${}_2{\mathbb V}_γ^β$ of all short exact sequences $$ \mathcal E: 0\to N_α\to N_β\to N_γ\to 0$$ where $α$ is any partition with each part at most 2. Due to the condition on $α$, the isomorphism type of a sequence $\mathcal E$ is given by an arc diagram $Δ$; denote by ${\mathbb V}_Δ$ the subset of ${}_2{\mathbb V}_γ^β$ of all sequences isomorphic to $\mathcal E$. Thus, the space ${}_2{\mathbb V}_γ^β$ carries a stratification given by the subsets of type ${\mathbb V}_Δ$. We compute the dimension of each stratum and show that the boundary of a stratum ${\mathbb V}_Δ$ consists exactly of those ${\mathbb V}_{Δ'}$ where $Δ'$ is obtained from $Δ$ by a non-empty sequence of arc moves of five possible types {\bf (A) -- (E)}. The case where all three partitions are fixed has been studied in [3] and [4]. There, arc moves of types {\bf (A) -- (D)} suffice to describe the boundary of a ${\mathbb V}_Δ$ in ${\mathbb V}_{α,γ}^β$. Our fifth move {\bf (E)}, "explosion", is needed to break up an arc into two poles to allow for changes in the partition $α$.

math.RT↗
arXiv2013

The existence of Hall polynomials for posets of finite prinjective type

We prove the existence of Hall polynomials for prinjective representations of finite partially ordered sets of finite prinjective type. In Section 4 we shortly discuss consequences of the existence of Hall polynomials, in particular, we are able to define a generic Ringel-Hall algebra for prinjective representations of posets of finite prinjective type.

math.RT↗
arXiv2013

Arc diagram varieties

Let $k$ be an algebraically closed field and $α$, $β$, $γ$ be partitions. An algebraic group acts on the constructible set of short exact sequences of nilpotent $k$-linear operators of Jordan types $α$, $β$, and $γ$, respectively; we are interested in the stratification given by the orbits in the case where all parts of $α$ are at most 2. Geometric properties of the degeneration relation are controlled by the combinatorics of arc diagrams. The extended bubble sort algorithm is used to construct chains of orbits such that subsequent strata have dimension difference equal to one.

math.RT↗
arXiv2009

On Lie algebras associated with representation directed algebras

Let $B$ be a representation-finite $\mathbb{C}$-algebra. The $\mathbb{Z}$-Lie algebra $L(B)$ associated with $B$ has been defined by Ch. Riedtmann. If $B$ is representation-directed there is another $\mathbb{Z}$-Lie algebra associated with $B$ defined by C. M. Ringel and denoted by $\CK(B)$. We prove that the Lie algebras $L(B)$ and $\CK(B)$ are isomorphic for any representation-directed $\mathbb{C}$-algebra $B$.

math.RT↗