Representations of Coxeter groups and homology of Coxeter graphs

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to the second-highest 2-sided cell in the sense of Kazhdan-Lusztig. We also give a description of the cell representation provided by this 2-sided cell, and find out all its simple quotients for simply laced Coxeter system with no more than one circuit in the Coxeter graph.Comment: 21 pages. Any comments welcome

Representations of Coxeter groups of Lusztig's a-function value 1

In this paper, we give a characterization of Coxeter group representations of Lusztig's a-function value 1, and determine all the irreducible such representations for certain simply laced Coxeter groups.Comment: This is a revised version of part of arXiv:2112.03771. 21 pages. Comments welcome

Generalized Vaidya Solutions and Misner-Sharp mass for nn-dimensional massive gravity

Dynamical solutions are always of interest to people in gravity theories. We derive a series of generalized Vaidya solutions in the $n$-dimensional de Rham-Gabadadze-Tolley (dRGT) massive gravity with a singular reference metric. Similar to the case of the Einstein gravity, the generalized Vaidya solution can describe shining/absorbing stars. Moreover, we also find a more general Vaidya-like solution by introducing a more generic matter field than the pure radiation in the original Vaidya spacetime. As a result, the above generalized Vaidya solution is naturally included in this Vaidya-like solution as a special case. We investigate the thermodynamics for this Vaidya-like spacetime by using the unified first law, and present the generalized Misner-Sharp mass. Our results show that the generalized Minser-Sharp mass does exist in this spacetime. In addition, the usual Clausius relation $\delta Q= TdS$ holds on the apparent horizon, which implicates that the massive gravity is in a thermodynamic equilibrium state. We find that the work density vanishes for the generalized Vaidya solution, while it appears in the more general Vaidya-like solution. Furthermore, the covariant generalized Minser-Sharp mass in the $n$-dimensional de Rham-Gabadadze-Tolley massive gravity is also derived by taking a general metric ansatz into account.Comment: 10 pages, no figure, version published in PR

Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality

Bj\"orner and Ekedahl [Ann. of Math. (2), 170.2(2009), pp. 799--817] pioneered the study of length-counting sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this paper, we study the asymptotic behavior of these sequences in affine Weyl groups. Let $W$ be an affine Weyl group with corresponding Weyl group $W_f$ and ${}^{f}{W}$ be the set of minimal representatives for the right cosets $W_f \backslash W$. Let $t_{\lambda}$ be the translation by a dominant coroot lattice element $\lambda$ and ${}^{f}{b}_i^{t_{\lambda}}$ be the number of elements of length $i$ below $t_\lambda$ in the Bruhat order on ${}^{f}{W}$. We show that the sequence $({}^{f}{b}_i^{t_{\lambda}})_i$ is ''asymptotically log-concave'' in the following sense: The sequence of discrete measures $(\mathfrak{m}_k)_k$ constructed from the $k$-fold dilated sequence $({}^{f}{b}_i^{t_{k\lambda}})_i$, as $k$ tends to infinity, converges weakly to a continuous measure obtained from a polytope $P^\lambda$. Moreover, the sequence of step functions $(S_k)_k$ of $({}^{f}{b}_i^{t_{k\lambda}})_i$ converges uniformly to the density function of this continuous measure. By Brunn--Minkowski inequality, this density is log-concave.Comment: 39 pages, 6 figures. The new result is stronger and includes uniform convergence. The introductory section has been vastly improve