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## On the algebra generated by projectors with commutator relation

In this article we apply methods of representation theory and combinatorial algebra to the different problems related to quantum tomography. For this purpose, we introduce the algebra generated by projectors satisfying some commutator relation. In this paper we study this commutator relation by combinatorialmethods and develop the representation theory of this algebra. Also, we apply our results to the case of mutually unbiased bases in dimension 7.

The goal of our arti le is a study of related mathemati al and physi al ob je ts: orthogonal pairs in sl(n) and mutually unbiased bases in Cn . An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respe t to the Killing form. The des ription of orthogonal pairs in a given Lie algebra is an important step in the lassi ation of orthogonal de- ompositions, i.e., de ompositions of the Lie algebra into a dire t sum of Cartan subalgebras pairwise orthogonal with respe t to the Killing form. One of the important notions of quantum me hani s, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert spa e C n . Two orthonormal bases {ei } n i=1 ; {fj } n j=1 are mutual ly unbiased if and only if |hei |fj i|2 = 1 n for any i; j = 1; : : : ; n. The notions of mutually unbiased bases in C n and orthogonal pairs in sl(n) are losely related. The problem of lassi ation of orthogonal pairs in sl(n) and the losely related problem of lassi ation of mutually unbiased bases in C n are still open even for the ase n = 6. In this arti le, we give a sket h of our proof that there is a omplex four-dimensional family of orthogonal pairs in sl(6). This proof requires a lot of algebrai geometry and representation theory. Further, we give an appli ation of the result on the algebrai geometri family to the study of mutually unbiased bases. We show the existen e of a real fourdimensional family of mutually unbiased bases in C 6 , thus solving a long-standing problem.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.