# American Institute of Mathematical Sciences

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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