Commuting nilpotent matrices and Artinian algebras

Basili, Roberta; Iarrobino, Anthony; Khatami, Leila

doi:10.1216/jca-2010-2-3-295

Cited by 16 publications

(51 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We denote this unique partition by Q(P ). The map P → Q(P ) has been studied by different authors (see [2], [3], [8], [9], [10], [11], [12]).…”

Section: Introductionmentioning

confidence: 99%

“…The number of parts of the partition Q(P ) was completely determined by R. Basili ([1,Proposition 2.4] and [3,Theorem 2.17]). It is known that if chark = 0 (see [10]) or char k > n (see [2]), then the map P → Q(P ) is idempotent: Q(Q(P )) = Q(P ).…”

Section: Introductionmentioning

confidence: 99%

“…CN C −1 ∈ U B (see [3,Lemma 2.2]). Thus to find the Jordan partition of a generic element of N B it is enough to study a generic element of U B .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

Khatami

2014

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

Let k be an infinite field. Fix a Jordan nilpotent n × n matrix B = J P with entries in k and associated Jordan type P . Let Q(P ) be the Jordan type of a generic nilpotent matrix commuting with B. In this paper, we use the combinatorics of a poset associated to the partition P , to give an explicit formula for the smallest part of Q(P ), which is independent of the characteristic of k. This, in particular, leads to a complete description of Q(P ) when it has at most three parts.Let k be an infinite field and Mat n (k) be the set of all n × n matrices with entries in k.Suppose that B = J P ∈ {Mat n (k)} is a Jordan matrix with associated Jordan type -Jordan block partition-P n. Recall that the centralizer and the nilpotent centralizer of MSC 2010: 05E40, 06A11, 14L30, 15A21

show abstract

“…We denote this unique partition by Q(P ). The map P → Q(P ) has been studied by different authors (see [2], [3], [8], [9], [10], [11], [12]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

Khatami

2014

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…. , v n according to decreasing i, then decreasing k, then decreasing u, as inFigure 2; see also[4,5,22].…”

mentioning

confidence: 96%

“…Summary of results on the Oblak Recursive Conjecture). Thus, the cases r P = 2[5,27,34,46] and r P = 3[26] of the Conjecture have been known since 2008 and 2012, respectively. Theorem 4.4 of the first and second authors then showed "half" the Conjecture in all characteristics.…”

mentioning

confidence: 99%

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

Iarrobino

Khatami

Steirteghem

et al. 2018

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions (P, Q), where Q = Q(P ) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P . T. Košir and P. Oblak have shown that Q has parts that differ pairwise by at least two. Such partitions, which are also known as "super distinct" or "Rogers-Ramanujan", are exactly those that are stable or "self-large" in the sense that Q(Q) = Q.In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q −1 (Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q −1 (Q) for Q = (u, u − r) with u > r > 1 could be arranged in an (r − 1) × (u − r) table T (Q) where the entry in the k-th row and -th column has k + parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q −1 (Q) for an arbitrary partition Q whose parts differ pairwise by at least two. *

show abstract

Strata of vector spaces of forms in R = k[x, y], and of rational curves in ℙ k

Iarrobino

2014

Bull Braz Math Soc, New Series

Self Cite

View full text Add to dashboard Cite

Consider the polynomial ring R = k[x, y] over an infinite field k and the subspace R j of degree-j homogeneous polynomials. The Grassmanian G = Grass(R j , d) parametrizes the vector spaces V ⊂ R j having dimension d. The strata Grass H (R j , d) ⊂ G determined by the Hilbert functions H = H(R/(V )) or, equivalently, by the Betti numbers of the algebras R/(V ), are locally closed and irreducible of known dimension. They satisfy a frontier property that the closure of a stratum is its union with lower strata [I1, I2]. The strata are determined also by the decomposition of the restricted tangent bundlewhere T is the tangent bundle on P d−1 and φ V is the rational curve determined by V . Each stratum corresponds to a partition, and the the poset of strata under closure is isomorphic to the poset of corresponding partitions in the Bruhat order. They are coarser than the strata defined by D. Cox, A. Kustin, C. Polini, and B. Ulrich [CKPU] and studied further in [KPU], that are determined in part by singularities of φ V . We explain these results and give examples to make them more accessible. We also generalize a result of D. Cox, T. Sederberg, and F. Chen and another of C. D'Andrea concerning the dimension and closure of µ families of parametrized rational curves from planar [CSC, D] to higher dimensional embeddings (Theorem 1.

show abstract

Commuting nilpotent matrices and Artinian algebras

Cited by 16 publications

References 15 publications

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

Strata of vector spaces of forms in R = k[x, y], and of rational curves in ℙ k

Contact Info

Product

Resources

About