Equivariant degenerations of spherical modules for groups of type A

Let G be a complex reductive algebraic group. Fix a Borel subgroup B of G, with unipotent radical U, and a maximal torus T in B with character group X(T). Let S be a submonoid of X(T) generated by finitely many dominant weights. V. Alexeev and M. Brion introduced a moduli scheme M_S which classifies pairs (X,f) where X is an affine G-variety and f is a T-equivariant isomorphism between the categorical quotient of X by U and the toric variety determined by S. In this paper, we prove that M_S is isomorphic to an affine space when S is the weight monoid of a spherical G-module with G of type A.Comment: v3: 65 pages, minor corrections and changes to exposition following referee's suggestions, a shorter version will appear in Annales de l'Institut Fourie

Asymptotic distribution of singular values of powers of random matrices

Let $x$ be a complex random variable such that {\E {x}=0}, {\E |x|^2=1}, {\E |x|^{4} < \infty}. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let {\Xb=(N^{-1/2}x_{ij})}, $1\leq i,j \leq N$ be a random matrix. Writing \Xb^* for the adjoint matrix of \Xb, consider the product \Xb^m{\Xb^*}^m with some $m \in \{1,2,...\}$. The matrix \Xb^m{\Xb^*}^m is Hermitian positive semi-definite. Let $\lambda_1,\lambda_2,...,\lambda_N$ be eigenvalues of \Xb^m{\Xb^*}^m (or squared singular values of the matrix \Xb^m). In this paper we find the asymptotic distribution function G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} of the empirical distribution function $${F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{\lambda_k \leq x\}}},$$ where $\mathbb{I} \{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy $$M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}.$$ In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967.Comment: 16 pages, 5 figure

Pro-Inflammatory Chemokines and Cytokines Dominate the Blister Fluid Molecular Signature in Patients with Epidermolysis Bullosa and Affect Leukocyte and Stem Cell Migration.

Hereditary epidermolysis bullosa (EB) is associated with skin blistering and the development of chronic nonhealing wounds. Although clinical studies have shown that cell-based therapies improve wound healing, the recruitment of therapeutic cells to blistering skin and to more advanced skin lesions remains a challenge. Here, we analyzed cytokines and chemokines in blister fluids of patients affected by dystrophic, junctional, and simplex EB. Our analysis revealed high levels of CXCR1, CXCR2, CCR2, and CCR4 ligands, particularly dominant in dystrophic and junctional EB. In vitro migration assays demonstrated the preferential recruitment of CCR4+ lymphocytes and CXCR1+, CXCR2+, and CCR2+ myeloid cells toward EB-derived blister fluids. Immunophenotyping of skin-infiltrating leukocytes confirmed substantial infiltration of EB-affected skin with resting (CD45RA+) and activated (CD45RO+) T cells and CXCR2+ CD11b+ cells, many of which were identified as CD16b+ neutrophils. Our studies also showed that abundance of CXCR2 ligand in blister fluids also creates a favorable milieu for the recruitment of the CXCR2+ stem cells, as validated by in vitro and in-matrix migration assays. Collectively, this study identified several chemotactic pathways that control the recruitment of leukocytes to the EB-associated skin lesions. These chemotactic axes could be explored for the refinement of the cutaneous homing of the therapeutic stem cells. © 2017 The Author