Hook young diagrams with applications to combinatorics and to representations of Lie superalgebras

We discuss existence of mixed stateAB of two (or multy-) com- ponent system HAB = HA HB with reduced density matricesA, �B and given spectraAB,�A,�B. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.

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Quantum marginal problem and representations of the symmetric group

Invariants and the ring of generic matrices

Codimensions of T-ideals and Hilbert series of relatively free algebras

We determine theSn×Sm-cocharacterXn,m of the algebraM1,1(E) and prove that theT2-ideal of its graded identities is generated by the polynomialsy1y2−y2y1 andz1z2z3+z3z2z1.

On the graded identities ofM1,1(E)

The Kronecker product of Sn-characters and an A ⊗ B theorem for Capelli identities

The multilinear identities of the k×k matrices F2 are studied in terms of the cocharacters and codimension sequences. For F2 the estimates are fairly good, while for k≥3 only partial results are obtained.

Amitai Regev

Papers

The Kronecker product of Sn-characters and an A ⊗ B theorem for Capelli identities

The polynomial identities of matrices in characteristic zero