2011, 5(2): 407-416. doi: 10.3934/amc.2011.5.407

The number of invariant subspaces under a linear operator on finite vector spaces

1. 

Institut für Mathematik, Karl-Franzens Universität Graz, Heinrichstr. 36/4, A-8010 Graz, Austria

Received  March 2010 Revised  April 2011 Published  May 2011

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F$q and $T$ a linear operator on $V$. For each $k\in\{1,\ldots,n\}$ we determine the number of $k$-dimensional $T$-invariant subspaces of $V$. Finally, this method is applied for the enumeration of all monomially nonisometric linear $(n,k)$-codes over $\mathbb F$q.
Citation: Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407
References:
[1]

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, "Error-Correcting Linear Codes - Classification by Isometry and Applications,'', Springer, (2006).

[2]

L. Brickman and P. A. Fillmore, The invariant subspace lattice of a linear transformation,, Can. J. Math., 19 (1967), 810. doi: 10.4153/CJM-1967-075-4.

[3]

H. Fripertinger, Enumeration of isometry classes of linear $(n,k)$-codes over $GF(q)$ in SYMMETRICA,, Bayreuth. Math. Schr., 49 (1995), 215.

[4]

H. Fripertinger, Enumeration of linear codes by applying methods from algebraic combinatorics,, Grazer Math. Ber., 328 (1996), 31.

[5]

H. Fripertinger, Cycle indices of linear, affine and projective groups,, Linear Algebra Appl., 263 (1997), 133. doi: 10.1016/S0024-3795(96)00530-7.

[6]

H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes,, in, (1995), 194.

[7]

, GAP Group,, GAP - Groups, (2008).

[8]

N. Jacobson, "Lectures In Abstract Algebra, II,'', D. Van Nostrand Company Inc., (1953).

[9]

A. Kerber, "Applied Finite Group Actions,'', Springer, (1999).

[10]

J. P. S. Kung, The cycle structure of a linear transformation over a finite field,, Linear Algebra Appl., 36 (1981), 141. doi: 10.1016/0024-3795(81)90227-5.

[11]

W. Lehmann, Das Abzähltheorem der Exponentialgruppe in gewichteter Form (in German),, Mitt. Math. Sem. Giessen, 112 (1974), 19.

[12]

W. Lehmann, "Ein vereinheitlichender Ansatz für die REDFIELD - PÓLYA - de BRUIJNSCHE Abzähltheorie,'', Ph.D thesis, (1976).

[13]

G. E. Séguin, The algebraic structure of codes invariant under a permutation,, in, (1996), 1.

[14]

D. Slepian, Some further theory of group codes,, Bell Sys. Techn. J., 39 (1960), 1219.

[15]

D. Slepian, Some further theory of group codes,, in, (1973), 118.

[16]

, SYMMETRICA,, A program system devoted to representation theory, (9544).

show all references

References:
[1]

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert and A. Wassermann, "Error-Correcting Linear Codes - Classification by Isometry and Applications,'', Springer, (2006).

[2]

L. Brickman and P. A. Fillmore, The invariant subspace lattice of a linear transformation,, Can. J. Math., 19 (1967), 810. doi: 10.4153/CJM-1967-075-4.

[3]

H. Fripertinger, Enumeration of isometry classes of linear $(n,k)$-codes over $GF(q)$ in SYMMETRICA,, Bayreuth. Math. Schr., 49 (1995), 215.

[4]

H. Fripertinger, Enumeration of linear codes by applying methods from algebraic combinatorics,, Grazer Math. Ber., 328 (1996), 31.

[5]

H. Fripertinger, Cycle indices of linear, affine and projective groups,, Linear Algebra Appl., 263 (1997), 133. doi: 10.1016/S0024-3795(96)00530-7.

[6]

H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes,, in, (1995), 194.

[7]

, GAP Group,, GAP - Groups, (2008).

[8]

N. Jacobson, "Lectures In Abstract Algebra, II,'', D. Van Nostrand Company Inc., (1953).

[9]

A. Kerber, "Applied Finite Group Actions,'', Springer, (1999).

[10]

J. P. S. Kung, The cycle structure of a linear transformation over a finite field,, Linear Algebra Appl., 36 (1981), 141. doi: 10.1016/0024-3795(81)90227-5.

[11]

W. Lehmann, Das Abzähltheorem der Exponentialgruppe in gewichteter Form (in German),, Mitt. Math. Sem. Giessen, 112 (1974), 19.

[12]

W. Lehmann, "Ein vereinheitlichender Ansatz für die REDFIELD - PÓLYA - de BRUIJNSCHE Abzähltheorie,'', Ph.D thesis, (1976).

[13]

G. E. Séguin, The algebraic structure of codes invariant under a permutation,, in, (1996), 1.

[14]

D. Slepian, Some further theory of group codes,, Bell Sys. Techn. J., 39 (1960), 1219.

[15]

D. Slepian, Some further theory of group codes,, in, (1973), 118.

[16]

, SYMMETRICA,, A program system devoted to representation theory, (9544).

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