We introduce a new notion of the determinant, called symmetrized determinant, for a square matrix with the entries in an associative algebra $Å$. The monomial expansion of the symmetrized determinant is obtained from the standard expansion of the commutative determinant by averaging the products of entries of the matrix in all possible orders. We show that for any fixed finite-dimensional associative algebra $Å$, the symmetrized determinant of an $n\times n$ matrix with the entries in $Å$ can be computed in polynomial in $n$ time (the degree of the polynomial is linear in the dimension of $Å$). Then, for every associative algebra $Å$ endowed with a scalar product and unbiased probability measure, we construct a randomized polynomial time algorithm to estimate the permanent of non-negative matrices. We conjecture that if $Å=\Mat(d, {\Bbb R})$ is the algebra of $d\times d$ real matrices endowed with the standard scalar product and Gaussian measure, the algorithm approximates the permanent of a non-negative $n \times n$ matrix within $O(γ_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} γ_d=1$. Finally, we provide some informal arguments why the conjecture might be true.
We view the determinant and permanent as functions on directed weighted graphs and introduce their analogues for the undirected graphs. We prove that the task of computing the undirected determinants as well as permanents for planar graphs, whose vertices have degree at most 4, is \#P-complete. In the case of planar graphs whose vertices have degree at most 3, the computation of the undirected determinant remains \#P-complete while the permanent can be reduced to the FKT algorithm, and therefore is polynomial. The undirected permanent is a Holant problem and its complexity can be deduced from …
We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.
The tensor rank and border rank of the $3 \times 3$ determinant tensor is known to be $5$ if characteristic is not two. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well. We also include an analysis of $5 \times 5$ and $7 \times 7$ determinant and permanent tensors, as well as the symmetric $3 \times 3$ permanent and determinant tensors. We end with some remarks on binary tensors.
In this paper, we prove a conjecture raised by Zhi-Wei Sun in 2018 by the eigenvectors-eigenvalues identity found by Denton, Parke, Tao and X. Zhang in 2019.
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n)=p(n). The proof uses symmetric functions.
The permanent-determinant method and its generalization, the Hafnian-Pfaffian method, are methods to enumerate perfect matchings of plane graphs that was discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanent-determinant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with 4n vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edg…
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of Bruhn and Rautenbach. We also obtain an upper bound on the number of perfect matchings in $C_4$-free bipartite graphs based on the number of edges, which, in the sparse case, improves on the classical Bregman's inequality for permanents. This bound is tight, as equality is achieved by the graph formed by vertex disjoint union of 6-vertex cycles.
Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant (resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by $a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property.
Let $\FF$ be a finite field of characteristics different from two. We show that no bijective map transforms permanent into determinant when the cardinality of $\FF$ is sufficiently large. We also give an example of non-bijective map when $\FF$ is arbitrary and an example of a bijective map when $\FF$ is infinite which do transform permanent into determinant. The developed technique allows us to estimate the probability of the permanent and the determinant of matrices over finite fields to have a given value. Our results are also true over finite rings without zero divisors.
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser. A 66 (1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the project of proving all ten formulas. We enumerate cyclically symmetric, self-complementary plane partitions. We first convert plane partitions to tilings of a hexagon in the plane by rhombuses, or equivalently to matchings in a certain planar graph. We can then use the permanent-determinant method …
Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a $β$-extension that is both a generalization of Sylvester's identity and the $β$-extension of the MacMahon master theorem.
Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued the possibility of an alternate, purely algebraic, polynomial-time approximation scheme. These schemes work by replacing each 1 with a random element of an algebra A, and considering the determinant of the resulting matrix. When A is noncommutative, this determinant can be defined in several ways. We show that for estimators based on the conventional determin…
Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets X,Y of size n, and determined its specialisation when Y and Z are the roots of y^n-a and z^n-b. We give the same specialisation for the determinant 1/\prod_x(xy-z), where {x} is an arbitrary set of indeterminates. The case of the Gaudin-Izergin-Korepin is for {x}={q,1/q}.
A determinantal approximation is obtained for the permanent of a doubly stochastic matrix. For moderate-deviation matrix sequences, the asymptotic relative error is of order $O(n^{-1})$.
In this paper we give the absolutely new proof of a conjecture of R.F.Scott(1881) on the permanent of a Cauchy matrix $\ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n-1$ and $y^n +1,$ respectively. The simple formula is given for the permanent of the Cauchy matrix $A= \ls \frac{1}{x_i-y_j} \rs_{1 \leqslant i,j \leqslant n},$ where $x_1, ..., x_n$ and $y_1, ..., y_n$ are the distinct roots of the polynomials $x^n+a$ and $y^n +b$, respectively: \begin{gather*} \per (A) =\frac{n}{(b-a)^n} \prod_{k=1…
We show that the permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers αand β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with a discussion of the computation…
In previous work, we showed that the solution of certain systems of discrete integrable equations, notably $Q$ and $T$-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras [BZ], the Kontsevich evolution [DFK09b] and the $T$…
We show that the weight enumerator of any binary linear code is equal to the permanent of a 3-dimensional hypermatrix (3-matrix). We also show that each permanent is a determinant of a 3-matrix. As an application we write the dimer partition function of a finite 3-dimensional cubic lattice as the determinant of the vertex-adjacency 3-matrix of a 2-dimensional simplicial complex which preserves the natural embedding of the cubic lattice.
We give new definitions for the determinant over commutative ring $K$, noncommutative ring $\mathbf{K}$, noncommutative ring $\mathcal{K}$ with associative powers, over noncommutative nonassociative ring $\mathfrak{K}$, and study their properties.
The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with $n$ crossings, we define a 7n by 7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time …
In this paper, we introduce a notion of quantum discrepancy, a non-commutative version of combinatorial discrepancy which is defined for projection systems, i.e. finite sets of orthogonal projections, as non-commutative counterparts of set systems. We show that besides its natural algebraic formulation, quantum discrepancy, when restricted to set systems, has a probabilistic interpretation in terms of determinantal processes. Determinantal processes are a family of point processes with a rich algebraic structure. A common feature of this family is the local repulsive behavior of points. Alisha…
The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutat…
Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint distribution of operators on a Fock space.
Expressions involving the product of the permanent with the (n-1)th power of the determinant of a matrix of indeterminates, and of (0,1)-matrices, are shown to be related to two conjectures that extend the Alon-Tarsi Latin square conjecture to odd dimensions. An identity involving an alternating sum of permanents of (0,1)-matrices is obtained.
Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $μ$, we define the polynomial $$P_μ(A)=\sum_{σ\in S_n} a_{1σ(1)}\cdots a_{nσ(n)}\,μ^{\ell(σ)}\; ,$$ as the $μ$-permanent of $A$, where $\ell(σ)$ is the number of inversions of the permutation $σ$ in the symmetric group $S_n$. In this note, we review several less known results of the $μ$-permanent, recalling some of its interesting properties. Some determinantal conjectures are considered and extended to that polynomial. A correction to a previous note is presented as well.
R. Redheffer described an $n\times n$ matrix of 0's and 1's the size of whose determinant is connected to the Riemann Hypothesis. We describe the permutations that contribute to its determinant and evaluate its permanent in terms of integer factorizations. We generalize the Redheffer matrix to finite posets that have a 0 element and find the analogous results in the more general situation.
I conjecture three identities for the determinant of adjacency matrices of graphene triangles and trapezia with Bloch (and more general) boundary conditions. For triangles, the parametric determinant is equal to the characteristic polynomial of the symmetric Pascal matrix. For trapezia it is equal to the determinant of a sub-matrix. Finally, the determinant of the tight binding matrix equals its permanent. The conjectures are supported by analytic evaluations and Mathematica, for moderate sizes. They establish connections with counting problems of partitions, lozenge tilings of hexagons, dense…
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision …
Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on…
The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we prove a conjecture concerning the minimality of Galois-Togliatti systems.
The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows: * The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible. * There exists an adjoint matrix $\adj{A}$ such that the matrix $A \adj{A}$ behaves much like the identity matrix (times $|A|$). * Every matrix $A$ is a supertropical root of its Hamilton-Cayley polynomial $f_A$. If these roots are distinct, then $A$ is conjugate (in a certain supertropical sense) to a diagonal matrix. * The tropical d…
