Geometric orbit datum and orbit covers

Vogan conjectured that the parabolic induction of orbit data is independent of the choice of the parabolic subgroup. In this paper we first give the parabolic induction of orbit covers, whose relationship with geometric orbit datum is also induced. Hence we show a geometric interpretation of orbit data and finally prove the conjugation for geometric orbit datum using geometric method.     

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck’s conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher’s bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.     

Induced orbit data and induced unitary representations for complex groups

We use induced orbit covers to define induced orbit data.By studying the space of regular functions on orbit cover,we know that the induced representation has close connection with the induced orbit datum under the meaning of Vogan’s conjecture.Therefore,when verifying Vogan’s conjecture,many cases can be reduced to the case of rigid orbit data.     

Comments on conjectures of trace inequalities on a generalized Wigner-Yanase skew information

Furuichi, Yanagi and Kuriyama gave three conjectures of trace inequalities on the Wigner-Yanase-Dyson skew information and a generalized Wigner-Yanase skew information (Furuichi et al. (2009) [1]) and Yanagi found a counterexample showing that two of the three conjectures don't hold (Yanagi (2010) [6]). In this note, we show that the last conjecture does not hold in general. In addition, we show that in the case of 2×2 matrices the conjecture is true.     

Geometric approach to parabolic induction

In this paper we construct a “restriction” map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on the parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig–Spaltenstein on induced unipotent classes to all infinite fields. We also prove a group version of a theorem of Harish-Chandra about the density of the span of regular semisimple orbital integrals.     

Stanley's character polynomials and coloured factorisations in the symmetric group

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.     

Independent coverings and orthogonal colourings

In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.     

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group that naturally arises investigating simple Heffter systems. Then we show its connection with related open problems and we present some results about the validity of these conjectures.     

Some remarks on Richardson orbits in complex symmetric spaces

Roger W. Richardson proved that any parabolic subgroup of a complex semisimple Lie group admits an open dense orbit in the nilradical of its corresponding parabolic subalgebra. In the case of complex symmetric spaces we show that there exist some large classes of parabolic subgroups for which the analogous statement which fails in general, is true. Our main contribution is the extension of a theorem of Peter E. Trapa (in 2005) to real semisimple exceptional Lie groups.

     

On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit

It is known that the equivariant diffeomorphism group Diff_G^∞(M)₀ of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. The purpose of this paper is to determine the first homology group of Diff_G^∞(M)₀ when M is a G-manifold with codimension one orbit.