Determinantal Identities Research Articles

On q-Isomonodromic Deformations and q-Nekrasov Functions

We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $\infty$, we obtain several determinantal identities of the tau functions.

Factorial Supersymmetric Skew Schur Functions and Ninth Variation Determinantal Identities

The determinantal identities of Hamel and Goulden have recently been shown to apply to a tableau-based ninth variation of skew Schur functions. Here we extend this approach and its results to the analogous tableau-based ninth variation of supersymmetric skew Schur functions. These tableaux are built on entries taken from an alphabet of unprimed and primed numbers and that may be ordered in a myriad of different ways, each leading to a determinantal identity. At the level of the ninth variation the corresponding determinantal identities are all distinct but the original notion of supersymmetry is lost. It is shown that this can be remedied at the level of the sixth variation involving a doubly infinite sequence of factorial parameters. Moreover, it is shown that the resulting factorial supersymmetric skew Schur functions are independent of the ordering of the unprimed and primed entries in the alphabet.

Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel’s Pfaffian outside decomposition identity. As corollaries to this we derive Pfaffian identities generalising those of Józefiak–Pragacz, Nimmo, and most recently Okada. As a preamble to this we present a parallel development based on (unshifted) semistandard tableaux that leads to a ninth variation version of the outside decomposition determinantal identity of Hamel and Goulden. In this case the corollaries we offer include determinantal identities generalising the Schur and skew Schur function identities of Jacobi–Trudi, Giambelli, Lascoux–Pragacz, Stembridge, and Okada.

Some determinantal considerations for pentadiagonal matrices

ABSTRACT In this note, we propose a research problem and two conjectures on the determinant of pentadiagonal matrices. At the same, we recall some results on pentadiagonal matrices and use them to provide new and short proofs of some recent determinantal identities for such matrices in the symmetric and skew-symmetric cases. This unified approach can be extended to further spectral consideration. In addition, two examples of explicit formulas of two non-trivial pentadiagonal matrices are provided.

Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’

Fibonacci-Like sequence associated with k-Pell, k-Pell-Lucas and Modified k-Pell sequences

The main aim of the present article to introduce a generalization of Fibonacci sequence which is similar to k-Pell, k-Pell-Lucas, Modified k-Pell sequences and known as Fibonacci-Like sequence. After that we obtain some fundamental properties of Fibonacci-Like sequence such as Binet formulae of Fibonacci-Like sequence, binomial transform of the Fibonacci-Like sequence and some other properties of Fibonacci-Like sequence with indexes in an arithmetic sequence. In addition to this we obtain some relations among k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences and the most worth noticing cause of the present paper is we present determinantal identities involving k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences.

Determinantal identities and examples of their applications

Determinantal identities and examples of their applications

A New Recurrence Relation and Related Determinantal form for Binomial Type Polynomial Sequences

For polynomial sequence of binomial type we propose a new recurrence formula equivalent to a determinantal form of the sequence, from which known and unknown properties are deducted. Relationships whit $${\delta}$$ –functionals, $${\delta}$$ –operators and generating function are pointed out. Finally, some classical examples are studied, highlighting interesting determinantal identities.

Determinantal identities for flagged Schur and Schubert polynomials

We prove new determinantal identities for a family of flagged Schur polynomials. As a corollary of these identities we obtain determinantal expressions of Schubert polynomials for certain vexillary permutations.

On some determinantal identities formation laws

We show that Muir’s law of extensible minors, Cayley’s law of complementaries and Jacobi’s identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769–791] are equivalent. We also show our generalization of Mühlbach/Muir’s extension principle [A generalization of Mühlbach’s extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363–1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach–Gasca–(Lopez-Carmona)–Ramirez identity [A generalization of Sylvester’s identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221–234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145–162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93–112].

Chiò's and Dodgson's determinantal identities

In this expository paper, we analyze and compare two determinantal identities constructed in the mid nineteenth century, Chiò's and Dodgson's, from the perspective of their origins in earlier work by J.J. Sylvester and C.G.J. Jacobi. All the known proofs of Chiò's and Dodgson's identities are cited and we present and compare them as modern algorithms for evaluating determinants. Restated as a recurrence, we discuss the role of Dodgson's identity in the development of the alternating sign matrix conjecture in the late twentieth century.

Extended Krein-Adler theorem for the translationally shape invariant potentials

Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Bäcklund transformations. It allows us to establish novel bilinear Wronskian and determinantal identities for classical orthogonal polynomials.

A generalization of Mühlbach's extension principle for determinantal identities

Mühlbach's extension principle for determinantal identities generalizes Muir's law of extensible minors. Some particular issues with Mühlbach–Beckermann's identity [A general determinantal identity of Sylvester type and some applications, Linear Algebra Appl. 197, 198 (1994), pp. 93–112] led to the conjecture of a more general extension method than Mühlbach's. However, no confirmation seems to have been reported so far. In this note, we present a generalization of Mühlbach's extension principle which confirms that conjecture. The whole identity of Mühlbach–Beckermann is put in a simpler form from which a new interpretation as an extension of Leibniz's definition of a determinant.

Determinants of matrices over semirings

In this paper, the concept of determinants for the matrices over a commutative semiring is introduced, and a development of determinantal identities is presented. This includes a generalization of the Laplace and Binet–Cauchy Theorems, as well as on adjoint matrices. Also, the determinants and the adjoint matrices over a commutative difference-ordered semiring are discussed and some inequalities for the determinants and for the adjoint matrices are obtained. The main results in this paper generalize the corresponding results for matrices over commutative rings, for fuzzy matrices, for lattice matrices and for incline matrices.

Explicit solutions for a (2 + 1)-dimensional Toda-like chain

We consider a (2 + 1)-dimensional Toda-like chain which can be viewed as a two-dimensional generalization of the Wu–Geng model and which is closely related to the two-dimensional Volterra, two-dimensional Toda and relativistic Toda lattices. In the framework of the Hirota direct approach, we present equations describing this model as a system of bilinear equations that belongs to the Ablowitz–Ladik hierarchy. Using the Jacobi-like determinantal identities and the Fay identity for the theta-functions, we derive its Toeplitz, dark-soliton and quasiperiodic solutions as well as the similar set of solutions for the two-dimensional Volterra chain.

Determinantal Identities for Generalized Fibonacci-Type Polynomials

In this Paper, we introduce Generalized Fibonacci-Type Polynomials. Further we present its determinantal identities with classical polynomials like Fibonacci Polynomial, Lucas Polynomials, Pell Polynomials, Pell-Lucas Polynomials, Cheby- shev polynomials of first and second kind.

Viewing Determinants as Nonintersecting Lattice Paths yields Classical Determinantal Identities Bijectively

In this paper, we show how general determinants may be viewed as generating functions of nonintersecting lattice paths, using the Lindström-Gessel-Viennot-method and the Jacobi Trudi identity together with elementary observations.After some preparations, this point of view provides "graphical proofs'' for classical determinantal identities like the Cauchy-Binet formula, Dodgson's condensation formula, the Plücker relations, Laplace's expansion and Turnbull's identity. Also, a determinantal identity generalizing Dodgson's condensation formula is presented, which might be new.

Cluster structures on quantum coordinate rings

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory $${\fancyscript{C}_{w}}$$ of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

In this paper, the confluent form of the multistep ɛ‐algorithm is proposed. The molecule solution of this system is derived by using determinantal identities. A new continuous prediction algorithm based on this confluent form is constructed. It also shows that this algorithm is a special case of the extended Lotka–Volterra system.

Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation

The molecule solution of an equation related to the lattice Boussinesq equation is derived with the help of determinantal identities. It is shown that this equation can for certain sequences be used as a numerical convergence acceleration algorithm. Numerical examples with applications of this algorithm are presented.