Asymptotics of the maximal and the typical dimensions of isotypic components of tensor representations of the symmetric group

Vershik and Kerov gave asymptotical bounds for the maximal and the typical dimensions of irreducible representations of symmetric groups $S_n$. It was conjectured by G. Olshanski that the maximal and the typical dimensions of the isotypic components of tensor representations of the symmetric group admit similar asymptotical bounds. The main result of this article is the proof of this conjecture. Consider the natural representation of $S_n$ on $(\mathbb{C}^N)^{\otimes n}$. Its isotypic components are parametrized by Young diagrams with $n$ cells and at most $N$ rows. P. Biane found the limit shape of Young diagrams when $n\rightarrow\infty,\ \sqrt{n}/N\rightarrow c$. By showing that this limit shape is the unique solution to a variational problem, it is proven here, that after scaling, the maximal and the typical dimensions of isotypic components lie between positive constants. A new proof of Biane's limit-shape theorem is obtained.Comment: To appear in European Journal of Combinatorics, special issue on "Groups, graphs and languages". 25 pages, 7 figures. The introduction and several sections were partially rewritte

Isoparametric foliation and Yau conjecture on the first eigenvalue

A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.Comment: to appear in J.Diff.Geo

Rigged Configurations and Kashiwara Operators

For types $A^{(1)}_n$ and $D^{(1)}_n$ we prove that the rigged configuration bijection intertwines the classical Kashiwara operators on tensor products of the arbitrary Kirillov-Reshetikhin crystals and the set of the rigged configurations.Comment: v2: 108 pages, the author's final version for publication, Proposition 33 added, Section 7.3 partially reworked; v3: published version (Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Fibers of flat morphisms and Weierstrass preparation theorem

We characterize flat extensions of commutative rings satisfying the Weierstrass preparation theorem. Using this characterization we prove a variant of the Weierstrass preparation theorem for rings of functions on a normal curve over a complete local domain of dimension one. This generalizes recent works of Harbater, Hartmann and Krashen with a different method of proof.Comment: To appear in Journal of Algebr

Isoparametric functions and exotic spheres

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an $S^3$-invariant isoparametric function on $Sp(2)$, we construct a properly transnormal but not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.Comment: 21 pages, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal

Associated primes of local cohomology modules and of Frobenius powers

We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the localization problem in tight closure theory

Cofiniteness of weakly Laskerian local cohomology modules

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm FD_{\leq n}}$ where $n\geq -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(or minimax) submodule N of $H^t_I(M)$ the R-modules ${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H^0_I(M)$, $H^1_I(M)$, ..., $H^{t-1}_I(M)$ are ${\rm FD_{\leq 1}}$ (or weakly Laskerian). As a consequence, it follows that the associated primes of $H^{t}_I(M)/N$ are finite. This generalizes the main results of Bahmanpour and Naghipour, Brodmann and Lashgari, Khashyarmanesh and Salarian, and Hong Quy. We also show that the category $\mathscr {FD}^1(R,I)_{cof}$ of $I$-cofinite ${\rm FD_{\leq1}}$ ~ $R$-modules forms an Abelian subcategory of the category of all $R$-modules.Comment: 8 pages, some changes has been don

Birationality of the tangent map for minimal rational curves

For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map.Comment: AMS-tex, 14 pages, Dedicated to Yum-Tong Siu on his 60th birthda