Combinatorial Representation Research Articles

The complex-type Pell p-numbers

In this paper, we define the complex-type Pell p-numbers and give the generating matrix of these defined numbers. Then, we produce the combinatorial representation, the generating function, the exponential representation and the sums of the complex-type Pell p-numbers. Also, we derive the determinantal and the permanental representations of the complex-type Pell p-numbers by using certain matrices which are obtained from the generating matrix of these numbers. Finally, we obtain the Binet formula for the complex-type Pell p-number.

Minimal generating sets and the abelianization for the quasitoric braid group

A toric braid is a braid whose closure is a torus link in [Formula: see text]. Manturov [A combinatorial representation of links by quasitoric braids, European J. Combin. 23(2) (2002) 207–212] generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization.

On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras

AbstractWe introduce the combinatorial model of J-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.

Cyclotomic Generating Functions

It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. These and related polynomials have been studied for many years by a variety of authors from the fields of combinatorics, representation theory, probability, number theory, and commutative algebra. We call such polynomials cyclotomic generating functions (CGFs). With Konvalinka, we have studied the support and asymptotic distribution of the coefficients of several families of CGFs arising from tableau and forest combinatorics. In this paper, we survey general CGFs from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGFs in combinatorial representation theory; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. As a sample result, we show that CGFs are "generically" asymptotically normal, generalizing a result of Diaconis on $q$-binomial coefficients using work of Hwang-Zacharovas. We include several open problems concerning CGFs.

Row–column duality and combinatorial topological strings

Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional topological field theories (TQFTs) based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

Solvable Stationary Non Equilibrium States

We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135–171, 2020). By combining duality and integrability the authors of (Frassek and Giardiná in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications.

New structure on the quantum alcove model with applications to representation theory and Schubert calculus

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant K -group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.

The Complex-Typek-Pell Numbers and Their Applications

In this study, a new sequence called the complex-typek-Pell number is defined. Also, we give properties of this sequence such as the generating matrix, the generating function, the combinatorial representations, the exponential representation, the sums, the permanental and determinantal representations, and the Binet formula. Then, we determine the periods of the recurrence sequence according to the moduloυand produce cyclic groups with the help of the generating matrices of the sequence. We also get some findings about the ranks and periods of the complex-typek-Pell sequence. Additionally, we create relations between the orders of the cyclic groups produced and the periods of the sequence. Then, this sequence is moved to groups and examined in detail in finite groups. As an application, we get the periods of the complex-type2-Pell numbers in the polyhedral groupsυ,2,2,2,υ,2, and2,2,υand the quaternion groupQ2υ.

Computational infrastructure for concepts discovery in science and technology

Automated scientific discovery, a topic in artificial intelligence has mainly been used to generate scientific insight from data. Our work follows the knowledge-driven discovery approach and introduces the use of category theory as the foundation for modeling diverse engineering fields represented with combinatorial representation. We show how category theory provides support for all stages of the discovery process starting from modeling the engineering knowledge. We demonstrate the use of the approach to rediscover previous discoveries in mechanics and discover new devices, some of which need to be realized to be appreciated. Category theory allows expanding the process to disciplines not modeled with combinatorial representations. We intend to demonstrate this in future studies.

On the Laplacian and Signless Laplacian Characteristic Polynomials of a Digraph

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L(D)=Deg+(D)−A(D) and Q(D)=Deg+(D)+A(D), where A(D) represents the adjacency matrix and Deg+(D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations.

The complex-type Fibonacci p-Sequences

In this paper, we define a new sequence which is called the complex-type Fibonacci p-sequence and we obtain the generating matrix of this complex-type Fibonacci p-sequence. We also derive the determinantal and the permanental representations. Then, using the roots of the characteristic polynomial of the complex-type Fibonacci p-sequence, we produce the Binet formula for this defined sequence. In addition, we give the combinatorial representations, the generating function, the exponential representation and the sums of the complex-type Fibonacci p-numbers.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including interactive proof systems, quantum information and computation, algorithmic coding theory, arithmetic complexity, expansion of hypergraphs and simplicial complexes, Markov chain Monte Carlo, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, extremal combinatorics, combinatorial number theory, probability theory, representation theory, and operator algebras.

Low-Complexity Detection for Multidimensional Codebooks Over Fading Channels

Signal space diversity (SSD) introduced by Boutros and Viterbo tremendously improves the error performance over fading channels without any power or bandwidth sacrifice. Maximum benefit is obtained with full diversity (FD) multidimensional codebooks and the rather complex maximum likelihood (ML) detection. In the present work, we propose a generalized combinatorial representation and an associated low-complexity list-based detection algorithm that works for any non-full or full diversity multidimensional codebook. The algorithm includes three options of lists offering different complexity-performance trade-offs. One of the lists has random size and is guaranteed to contain the ML point, while the other two lists have fixed or limited size. Some analytical results are presented for the list sizes. The algorithm also includes a smart search to find the most probable codebook point in the list. Numerical results show that the proposed algorithm yields optimal or close-to-optimum performance with a noteworthy detection complexity reduction. Therefore, its adoption can be of practical interest, especially in applications where powerful error-correcting codes are not supported.

On the coefficients of skew Laplacian characteristic polynomial of digraphs

Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see text] edges. Let [Formula: see text] be the digraph obtained by orienting the edges of [Formula: see text] arbitrarily. The digraph [Formula: see text] is called an orientation of [Formula: see text] or oriented graph corresponding to [Formula: see text]. The skew Laplacian matrix of the digraph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] is the skew matrix and [Formula: see text] is the diagonal matrix with [Formula: see text]th diagonal entry [Formula: see text]. In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of [Formula: see text]. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determinations of the directed path and directed cycle.

Geometric Nature of Relations on Plabic Graphs and Totally Non-negative Grassmannians

Abstract The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicoloured (plabic) graphs in the disc. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations at the vertices of such graphs, depending on some signatures defined on their edges. The problem of characterizing the signatures corresponding to the totally non-negative cells was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non-negativity property on the full positroid cell. If each edge in a graph $\mathcal G$ belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and the gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of the geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property in the case of bipartite graphs [1, 7], and has a different statistical mechanical interpretation otherwise [6]. An explicit connection between the solution of Lam’s system of relations and the value of Postnikov’s boundary measurement map is established using the generalization of Talaska’s formula [51] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for the transformations of the signatures under Postnikov’s moves and reductions and amalgamations of networks.

A Discrete Representation of the Second Fundamental Form

We present a new method to obtain a combinatorial representation for the behaviour of a submanifold isometrically immersed in a Riemannian manifold based on the second fundamental form. We also present several results applying this new way of representation.

The Hadamard-type k-step Pell sequences

In this paper, we define the Hadamard-type k-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the k-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type k-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type k-step Pell numbers for the case that k is odd integers and k ≥ 3. Finally, we derive some properties of the Hadamard-type k-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.

Gröbner geometry of Schubert polynomials through ice

The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gröbner degeneration of matrix Schubert varieties. We consider instead diagonal Gröbner degenerations. In this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatonarics for the class of “vexillary” matrix Schubert varieties. We initiate a study of general diagonal degenerations, relating them to a neglected formula of Lascoux (2002) in terms of the 6-vertex ice model (recently rediscovered by Lam, Lee, and Shimozono (2021) in the guise of “bumpless pipe dreams”).

The complex-type Padovan-p sequences

In this paper, we define the complex-type Padovan-p sequence and then give the relationships between the Padovan-p numbers and the complex-type Padovan-p numbers. Also, we provide a new Binet formula and a new combinatorial representation of the complex-type Padovan-p numbers by the aid of the nth power of the generating matrix of the complex-type Padovan-p sequence. In addition, we derive various properties of the complex-type Padovan-p numbers such as the permanental, determinantal and exponential representations and the finite sums by matrix methods.

On the co-complex-type [formula omitted]-Fibonacci numbers

In this paper, we define the co-complex-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the co-complex-type k-Fibonacci numbers. Also, we produce various properties of the co-complex-type k-Fibonacci numbers such as the generating matrices, the Binet formulas, the combinatorial, permanental and determinantal representations, and the finite sums by matrix methods. In addition, we study the co-complex-type k-Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the co-complex-type k-Fibonacci sequences for any k and m. Furthermore, we extend the co-complex-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the co-complex-type 2-Fibonacci sequences in the semidihedral group SD2m, (m≥4) with respect to the generating pair (x,y).