Group actions and invariants in algebras of generic matrices

Z. Reichstein; N. Vonessen

doi:10.1016/j.aam.2005.08.007

Group actions and invariants in algebras of generic matrices

Z. Reichstein
, N. Vonessen

Mathematical Sciences

University of British Columbia

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

2 Scopus citations

Abstract

We show that the fixed elements for the natural GL_m-action on the universal division algebra UD (m, n) of m generic n × n-matrices form a division subalgebra of degree n, assuming n ≥ 3 and 2 ≤ m ≤ n² - 2. This allows us to describe the asymptotic behavior of the dimension of the space of SL_m-invariant homogeneous central polynomials p (X₁, ..., X_m) for n × n-matrices. Here the base field is assumed to be of characteristic zero.

Original language	English
Pages (from-to)	481-500
Number of pages	20
Journal	Advances in Applied Mathematics
Volume	37
Issue number	4
DOIs	https://doi.org/10.1016/j.aam.2005.08.007
State	Published - Oct 2006

Funding

Keywords: Generic matrices; Universal division algebra; Central polynomial; PI-degree; Group action; Geometric action; Invariants; Concomitants; Gelfand–Kirillov dimension * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Z. Reichstein was supported in part by an NSERC research grant. 2 N. Vonessen gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

Funders
University of British Columbia

Keywords

Central polynomial
Concomitants
Gelfand-Kirillov dimension
Generic matrices
Geometric action
Group action
Invariants
PI-degree
Universal division algebra

Access to Document

10.1016/j.aam.2005.08.007

Cite this

@article{09903c713c064798b445d9d64ed8120f,

title = "Group actions and invariants in algebras of generic matrices",

abstract = "We show that the fixed elements for the natural GLm-action on the universal division algebra UD (m, n) of m generic n × n-matrices form a division subalgebra of degree n, assuming n ≥ 3 and 2 ≤ m ≤ n2 - 2. This allows us to describe the asymptotic behavior of the dimension of the space of SLm-invariant homogeneous central polynomials p (X1, ..., Xm) for n × n-matrices. Here the base field is assumed to be of characteristic zero.",

keywords = "Central polynomial, Concomitants, Gelfand-Kirillov dimension, Generic matrices, Geometric action, Group action, Invariants, PI-degree, Universal division algebra",

author = "Z. Reichstein and N. Vonessen",

note = "Funding Information: Keywords: Generic matrices; Universal division algebra; Central polynomial; PI-degree; Group action; Geometric action; Invariants; Concomitants; Gelfand–Kirillov dimension * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/\textasciitilde{}reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Z. Reichstein was supported in part by an NSERC research grant. 2 N. Vonessen gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.",

year = "2006",

month = oct,

doi = "10.1016/j.aam.2005.08.007",

language = "English",

volume = "37",

pages = "481--500",

journal = "Advances in Applied Mathematics",

issn = "0196-8858",

publisher = "Academic Press Inc.",

number = "4",

}

TY - JOUR

T1 - Group actions and invariants in algebras of generic matrices

AU - Reichstein, Z.

AU - Vonessen, N.

N1 - Funding Information: Keywords: Generic matrices; Universal division algebra; Central polynomial; PI-degree; Group action; Geometric action; Invariants; Concomitants; Gelfand–Kirillov dimension * Corresponding author. E-mail addresses: [email protected] (Z. Reichstein), [email protected] (N. Vonessen). URLs: http://www.math.ubc.ca/~reichst (Z. Reichstein), http://www.math.umt.edu/vonessen (N. Vonessen). 1 Z. Reichstein was supported in part by an NSERC research grant. 2 N. Vonessen gratefully acknowledges the support of the University of Montana and the hospitality of the University of British Columbia during his sabbatical in 2002/2003, when part of this research was done.

PY - 2006/10

Y1 - 2006/10

N2 - We show that the fixed elements for the natural GLm-action on the universal division algebra UD (m, n) of m generic n × n-matrices form a division subalgebra of degree n, assuming n ≥ 3 and 2 ≤ m ≤ n2 - 2. This allows us to describe the asymptotic behavior of the dimension of the space of SLm-invariant homogeneous central polynomials p (X1, ..., Xm) for n × n-matrices. Here the base field is assumed to be of characteristic zero.

AB - We show that the fixed elements for the natural GLm-action on the universal division algebra UD (m, n) of m generic n × n-matrices form a division subalgebra of degree n, assuming n ≥ 3 and 2 ≤ m ≤ n2 - 2. This allows us to describe the asymptotic behavior of the dimension of the space of SLm-invariant homogeneous central polynomials p (X1, ..., Xm) for n × n-matrices. Here the base field is assumed to be of characteristic zero.

KW - Central polynomial

KW - Concomitants

KW - Gelfand-Kirillov dimension

KW - Generic matrices

KW - Geometric action

KW - Group action

KW - Invariants

KW - PI-degree

KW - Universal division algebra

UR - https://www.scopus.com/pages/publications/33749644770

U2 - 10.1016/j.aam.2005.08.007

DO - 10.1016/j.aam.2005.08.007

M3 - Article

AN - SCOPUS:33749644770

SN - 0196-8858

VL - 37

SP - 481

EP - 500

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

IS - 4

ER -

Group actions and invariants in algebras of generic matrices

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this