Advanced Determinant Calculus

Abstract

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

67 pages, AmS-LaTeX; Final journal version. Some further, minor modifications

Cited by in corpus (25)

• On the weighted enumeration of alternating sign matrices and descending plane partitions
• A framework of Rogers-Ramanujan identities and their arithmetic properties
• Proof of George Andrews's and David Robbins's q-TSPP Conjecture
• Planar Linkages Following a Prescribed Motion
• A multi-dimensional Markov chain and the Meixner ensemble
• A new transform for solving the noisy complex exponentials approximation problem
• Multiply-refined enumeration of alternating sign matrices
• Determinantal Martingales and Noncolliding Diffusion Processes
• Discrete analogues of Macdonald-Mehta integrals
• Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
• AFLT-type Selberg integrals
• Nonlinear Random Matrix Statistics, symmetric functions and hyperdeterminants
• Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz
• The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants
• Identities from representation theory
• Skew Howe duality and limit shapes of Young diagrams
• Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities
• The Determinant of an Elliptic Sylvesteresque Matrix
• Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix
• A Curious Family of Binomial Determinants That Count Rhombus Tilings of a Holey Hexagon
• Hermite-Thue equation: Padé approximations and Siegel's lemma
• Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids
• Excursion Processes Associated with Elliptic Combinatorics
• The Bhargava greedoid as a Gaussian elimination greedoid
• Continued fractions for $q$-deformed real numbers, $\{-1,0,1\}$-Hankel determinants, and Somos-Gale-Robinson sequences