| Title:
|
Irreducible algebraic sets of matrices with dominant restriction of the characteristic map (English) |
| Author:
|
Skrzyński, Marcin |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
128 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
91-101 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We collect certain useful lemmas concerning the characteristic map, ${\mathcal GL}_n$-invariant sets of matrices, and the relative codimension. We provide a characterization of rank varieties in terms of the characteristic map as well as some necessary and some sufficient conditions for linear subspaces to allow the dominant restriction of the characteristic map. (English) |
| Keyword:
|
characteristic map |
| Keyword:
|
dominant map |
| Keyword:
|
linear subspace |
| Keyword:
|
$\mathcal G\mathcal L_n$-invariant set of matrices |
| Keyword:
|
rank variety |
| MSC:
|
13A50 |
| MSC:
|
14A10 |
| MSC:
|
14L30 |
| MSC:
|
15A03 |
| MSC:
|
15A18 |
| idZBL:
|
Zbl 1016.15005 |
| idMR:
|
MR1974548 |
| DOI:
|
10.21136/MB.2003.133938 |
| . |
| Date available:
|
2009-09-24T22:07:20Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133938 |
| . |
| Reference:
|
[1] Eisenbud, D., Saltman, D.: Rank varieties of matrices.Commutative Algebra (Berkeley, CA). Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989, pp. 173–212. MR 1015518 |
| Reference:
|
[2] Gantmacher, F. R.: Théorie des matrices.Dunod, Paris, 1966. Zbl 0136.00410 |
| Reference:
|
[3] Helton, W., Rosenthal, J., Wang, X.: Matrix extensions and eigenvalue completions, the generic case.Trans. Amer. Math. Soc. 349 (1997), 3401–3408. MR 1432201, 10.1090/S0002-9947-97-01975-2 |
| Reference:
|
[4] Lang, S.: Algebra.W. A. Benjamin, Inc., New York, 1970. Zbl 0216.06001 |
| Reference:
|
[5] Łojasiewicz, St.: Introduction to Complex Analytic Geometry.Birkhäuser, Basel, 1991. MR 1131081 |
| Reference:
|
[6] Shafarevich, I. R.: Basic Algebraic Geometry.Springer, Berlin, 1977. Zbl 0362.14001, MR 0447223 |
| Reference:
|
[7] Skrzyński, M.: Remarks on applications of rank functions to algebraic sets of matrices.Demonstratio Math. 32 (1999), 263–271. MR 1710249 |
| Reference:
|
[8] Skrzyński, M.: On ${\mathcal GL}_n$-invariant cones of matrices with small stable ranks.Demonstratio Math. 33 (2000), 243–254. MR 1769417 |
| Reference:
|
[9] Skrzyński, M.: On ${\mathcal GL}_n$-invariant algebraic cones of matrices with relative codimension equal to $1$.Commentationes Math. 40 (2000), 167–174. MR 1810393 |
| Reference:
|
[10] Weyman, J.: The equations of conjugacy classes of nilpotent matrices.Invent. Math. 98 (1989), 229–245. Zbl 0717.20033, MR 1016262, 10.1007/BF01388851 |
| . |
