It is known that the nilpotent commutator of a nilpotent matrix is an irreducible variety and that there is a unique partition such that the intersection of the orbit of nilpotent matrices corresponding to this partition is dense. This defines a map on the set of partitions. We show that this map is idempotent. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we show that a generic pair of commuting nilpotent matrices generates a Gorenstein algebra.
COBISS.SI-ID: 15077977
Description: Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB-BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This result is a common generalization of the results from [H. Radjavi, P. Rosenthal, J. Funct. Anal. 147 (1997) 443-456] and [G. Cigler, et alt., J. Funct. Anal. 160 (1998) 452-465]
COBISS.SI-ID: 15167321
We prove that the nonabelian tensor square of a powerful $p$-group is again a powerful $p$-group. Furthermore, if $G$ ispowerful, then the exponent of $G \otimes G$ divides the exponent of $G$. New bounds for the exponent, rank, and order of various homological functors of a given finite $p$-group are obtained. In particular, we improve the bound for the order of the Schur multiplier of a given finite $p$-group obtained by Lubotzky and Mann.
COBISS.SI-ID: 15456089
Given a reductive group G acting on an affine R-variety V, consider the induced dual action on the ring R[V] and on its dual. Given an invariant closed semialgebraic subset K of V(R), we study the representation of invariant nonnegative polynomials on K by invariant sums of squares, and representation of invariant linear functionals on R[V] by invariant measures. We analyse the relation between quadratic modules of R[V] and associated quadratic modules of the subring R[V]^G of invariant polynomials. We study finite solvability of a version of the multidimensional K-moment problem.
COBISS.SI-ID: 15111769
We analyze problems involving matrix variables using NC analytic functions, defined by power series in noncommuting variables, and evaluate these functions on sets of matrices of all dimensions. This approach has recent applications in linear system engineering. We characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary, called NC ball maps. Up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map. So, NC ball maps are simpler than the classical ones of D'Angelo.
COBISS.SI-ID: 15190361