Root systems and generalized associahedra 1 Root systems and generalized associahedra 3

Abstract. In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras. Contents

Abstract. We consider the class of totally nonnegative (TN) matrices—matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices. We present new O(n 3) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix. We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy. We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries. In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy–Vandermonde, and generalized Vandermonde matrices to high relative accuracy.

Abstract. Vandermonde, Cauchy, and Cauchy–Vandermonde totally positive linear systems can be solved extremely accurately in O(n 2) time using Björck–Pereyra-type methods. We prove that Björck–Pereyra-type methods exist not only for the above linear systems but also for any totally positive linear system as long as the initial minors (i.e., contiguous minors that include the first row or column) can be computed accurately. Using this result we design a new O(n 2) Björck– Pereyra-type method for solving generalized Vandermonde systems of equations by using a new algorithm for computing the Schur function. We present explicit formulas for the entries of the bidiagonal decomposition, the LDU decomposition, and the inverse of a totally positive generalized Vandermonde matrix, as well as algorithms for computing these entries to high relative accuracy.

Abstract. We consider the problem of performing accurate computations with rectangular (m × n) totally nonnegative matrices. The matrices under consideration have the property of having a unique representation as products of nonnegative bidiagonal matrices. Given that representation, one can compute the inverse, LDU decomposition, eigenvalues, and SVD of a totally nonnegative matrix to high relative accuracy in O(max(m 3,n 3)) time—much more accurately than conventional algorithms that ignore that structure. The contribution of this paper is to show that the high relative accuracy is preserved by operations that preserve the total nonnegativity—taking a product, re-signed inverse (when m = n), converse, Schur complement, or submatrix of a totally nonnegative matrix, any of which costs at most O(max(m 3,n 3)). In other words, the class of totally nonnegative matrices for which we can do numerical linear algebra very accurately in O(max(m 3,n 3)) time (namely, those for which we have a product representation via nonnegative bidiagonals) is closed under the operations listed above.

Abstract. This is an expanded version of the notes of our lectures

We show that the set of totally positive unipotent upper-triangular Toeplitz matrices in GLn form a real semi-algebraic cell of dimension n − 1. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of GLn(C) relying in particular on the positivity of the structure constants (Gromov–Witten invariants). We also give a characterization of total positivity for Toeplitz matrices in terms of (quantum) Schubert bases. This work builds on some results of Dale Peterson’s which we explain with proofs in the type A case.

1.1 Summary of results In this paper we study the connection between dimers and Harnack curves discovered in [12]. To any periodic edge-weighted planar bipartite graph Γ

Abstract. Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = AkAk−1 ···A1. Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of B T B, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.

In [7] a combinatorial criterion for quasi-commutativity is established for pairs of quantum Plücker coordinates in the quantized coordinate algebra Cq[F] of the flag variety of type A. This paper attempts to generalize these results by producing necessary and sufficient conditions for pairs of quantums minors in the quantized coordinate algebra Cq[Matk×m] to quasi-commute. In addition we study the combinatorics of maximal (by inclusion) families of pairwise quasi-commuting quantum minors and pose relevant conjectures. 1.