Arcs on Determinantal Varieties

We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute the number of irreducible components of jet schemes, log canonical thresholds, and topological zeta functions.Comment: 27 pages. This is part of the author's PhD thesis at the University of Illinois at Chicago. v2: Minor changes. To appear in Transactions of the American Mathematical Societ

Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Br\"and\'en has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Br\"and\'en's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence

Double Bruhat cells and total positivity

We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.Comment: 45 pages, 8 figures; for color version, see http://www-math.mit.edu/~fomin/papers.htm

Young-Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

We propose a new approach to a unified study of determinants, permanents, immanants, (determinantal) bitableaux and symmetrized bitableaux in the polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the enveloping algebra $U(gl(n))$. This leads to new relevant classes of elements in $U(gl(n))$: Capelli bitableaux, right Young-Capelli bitableaux and Capelli immanants. The set of standard Capelli bitableaux and the set of standard right Young-Capelli bitableaux are bases of $U(gl(n))$, whose action on the Gordan-Capelli basis of $C[M_{n, n}]$ have remarkable properties. Capelli immanants can be efficiently computed and provide a system of generators of $U(gl(n))$. The Okounkov quantum immanants are proved to be simple linear combinations of Capelli immanants. Several examples are provided throughout the paper.Comment: arXiv admin note: text overlap with arXiv:1608.0678