Rook placements and Jordan forms of upper-triangular nilpotent matrices

Martha Yip

doi:10.37236/6888

Rook placements and Jordan forms of upper-triangular nilpotent matrices

Martha Yip

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number F_λ (q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young’s lattice and non-attacking rook placements, which leads to a refinement of the formula for F_λ (q).

Original language	English
Article number	#P1.68
Journal	Electronic Journal of Combinatorics
Volume	25
Issue number	1
DOIs	https://doi.org/10.37236/6888
State	Published - Mar 29 2018

Bibliographical note

Publisher Copyright:
© 2018, Australian National University. All rights reserved.

Keywords

Finite fields
Jordan form
Nilpotent matrices
Rook placements
Set partitions
Young tableaux

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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10.37236/6888

Cite this

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AB - The set of n by n upper-triangular nilpotent matrices with entries in a finite field Fq has Jordan canonical forms indexed by partitions λ ⊢ n. We present a combinatorial formula for computing the number Fλ (q) of matrices of Jordan type λ as a weighted sum over standard Young tableaux. We construct a bijection between paths in a modified version of Young’s lattice and non-attacking rook placements, which leads to a refinement of the formula for Fλ (q).

KW - Finite fields

KW - Jordan form

KW - Nilpotent matrices

KW - Rook placements

KW - Set partitions

KW - Young tableaux

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U2 - 10.37236/6888

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ER -

Rook placements and Jordan forms of upper-triangular nilpotent matrices

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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