Combinatorial Basis Research Articles

Linear independence for [formula omitted] by using [formula omitted]

In this note we prove linear independence of the combinatorial spanning set for standard Cℓ(1)-module L(kΛ0) by establishing a connection with the combinatorial basis of Feigin-Stoyanovsky's type subspace W(kΛ0) of C2ℓ(1)-module L(kΛ0). It should be noted that the proof of linear independence for the basis of W(kΛ0) is obtained by using simple currents and intertwining operators in the vertex operator algebra L(kΛ0).

Stochastic Process Leading to Catalan Number Recurrence

Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of {1,2,…,N}, are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system.

The quantum detection of projectors in finite-dimensional algebras and holography

We define the computational task of detecting projectors in finite dimensional associative algebras with a combinatorial basis, labelled by representation theory data, using combinatorial central elements in the algebra. In the first example, the projectors belong to the centre of a symmetric group algebra and are labelled by Young diagrams with a fixed number of boxes n. We describe a quantum algorithm for the task based on quantum phase estimation (QPE) and obtain estimates of the complexity as a function of n. We compare to a classical algorithm related to the projector identification problem by the AdS/CFT correspondence. This gives a concrete proof of concept for classical/quantum comparisons of the complexity of a detection task, based in holographic correspondences. A second example involves projectors labelled by triples of Young diagrams, all having n boxes, with non-vanishing Kronecker coefficient. The task takes as input the projector, and consists of identifying the triple of Young diagrams. In both of the above cases the standard QPE complexities are polynomial in n. A third example of quantum projector detection involves projectors labelled by a triple of Young diagrams, with m, n and m + n boxes respectively, such that the associated Littlewood-Richardson coefficient is non-zero. The projector detection task is to identify the triple of Young diagrams associated with the projector which is given as input. This is motivated by a two-matrix model, related via the AdS/CFT correspondence, to systems of strings attached to giant gravitons. The QPE complexity in this case is polynomial in m and n.

Combinatorial bases of standard modules of twisted affine Lie algebras in types and : rectangular highest weights

We consider the standard modules of rectangular highest weights of affine Lie algebras in types and . By using vertex algebraic techniques we construct the combinatorial bases for standard modules and their principal subspaces and parafermionic spaces. Finally, we compute the corresponding character formulae and, as an application, we obtain two new families of combinatorial identities.

Matrix and tensor witnesses of hidden symmetry algebras

Permutation group algebras, and their generalizations called permutation centralizer algebras (PCAs), play a central role as hidden symmetries in the combinatorics of large N gauge theories and matrix models with manifest continuous gauge symmetries. Polynomial functions invariant under the manifest symmetries are the observables of interest and have applications in AdS/CFT. We compute such correlators in the presence of matrix or tensor witnesses, which by definition, can include a matrix or tensor field appearing as a coupling in the action (i.e a spurion) or as a classical (un-integrated) field in the observables, appearing alongside quantum (integrated) fields. In both matrix and tensor cases we find that two-point correlators of general gauge-invariant observables can be written in terms of gauge invariant functions of the witness fields, with coefficients given by structure constants of the associated PCAs. Fourier transformation on the relevant PCAs, relates combinatorial bases to representation theoretic bases. The representation theory basis elements obey orthogonality results for the two-point correlators which generalise known orthogonality relations to the case with witness fields. The new orthogonality equations involve two representation basis elements for observables as input and a representation basis observable constructed purely from witness fields as the output. These equations extend known equations in the super-integrability programme initiated by Mironov and Morozov, and are a direct physical realization of the Wedderburn-Artin decompositions of the hidden permutation centralizer algebras of matrix/tensor models.

Diamond Cone for $${{\mathfrak {sl}}}(m,n)$$

In this paper, we study the shape and the reduced shape algebra for the Lie superalgebra \({\mathfrak {sl}}(m,n)\). We define quasistandard Young tableaux, and they give a combinatorial basis for the reduced shape algebra, called the diamond cone. There is a natural bijective map from the set of semistandard tableaux with shape \(\lambda \) to the set of quasistandard tableaux with smaller shape: the diamond cone is compatible with the stratification of the reduced shape algebra.

An Analysis of TENZI Using Combinatorics and Markov Chains

Summary TENZI, “the world’s fastest game,” sounds too simple: roll 10 dice until they all match, setting aside the dice you want to keep. How fast is it? That is, on average how many rolls does it take to get 10 matches? An intuitive strategy that keeps the largest group (LG) of matching dice after each roll turns out to be optimal. The average number of rolls, for both the optimal and a naive policy, are found recursively from basic combinatorics, with some ideas from order statistics and Markov chains. The LG strategy is shown to be optimal by formulating TENZI as a dynamic program, showing that it can be reduced to a much smaller problem, and numerically solving this problem to obtain the optimal policy.

Parafermionic bases of standard modules for affine Lie algebras

In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra $\widehat{\mathfrak sl}(n+1,\mathbb C)$---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-laced untwisted affine Lie algebras.

Learning to Solve Counting Problems: Challenges and Opportunities for Non-Math Majors

Our study stems from the belief that meaningful and challenging mathematics can be made accessible to all students. To this end, we designed and implemented a special course for undergraduate non-mathematics and non-mathematics education majors. The course entitled Counting and Chance satisfied the requirement of a core course in quantitative reasoning. The content chosen as the focus of this course was basic combinatorics (i.e., counting problems) – a topic that is non-procedural in nature and could be made accessible to students with limited mathematical background. The design of the course was inspired by inclusive pedagogical principles that motivate and support students’ conceptual learning. We present the characteristics of the learning environment that was developed within the framework of this course, point to the rich and sophisticated kinds of reasoning that students developed throughout the course, and discuss how students’ collaborative engagement in the learning process seemed to have contributed to their views of themselves as math learners and their conceptions of what math is. Our findings point to the feasibility of such a course for non-math students, and its potential merit in helping students: (i) develop appreciation of mathematics as a topic that requires thinking, reasoning, and convincing; (ii) become more confident in their ability to do math.

Combinatorial constructions of derived equivalences

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space.

Expressivity of the key genes associated with seed and pod development is highly regulated via lncRNAs and miRNAs in Pigeonpea

Non-coding RNA’s like miRNA, lncRNA, have gained immense importance as a significant regulatory factor in different physiological and developmental processes in plants. In an effort to understand the molecular role of these regulatory agents, in the present study, 3019 lncRNAs and 227 miRNAs were identified from different seed and pod developmental stages in Pigeonpea, a major grain legume of Southeast Asia and Africa. Target analysis revealed that 3768 mRNAs, including 83 TFs were targeted by lncRNAs; whereas 3060 mRNA, including 154 TFs, were targeted by miRNAs. The targeted transcription factors majorly belong to WRKY, MYB, bHLH, etc. families; whereas the targeted genes were associated with the embryo, seed, and flower development. Total 302 lncRNAs interact with miRNAs and formed endogenous target mimics (eTMs) which leads to sequestering of the miRNAs present in the cell. Expression analysis showed that notably, Cc_lncRNA-2830 expression is up-regulated and sequestrates miR160h in pod leading to higher expression of the miR160h target gene, Auxin responsive factor-18. A similar pattern was observed for SPIKE, Auxin signaling F-box-2, Bidirectional sugar transporter, and Starch synthetase-2 eTMs. All the identified target mRNAs code for transcription factor and genes are involved in the processes like cell division, plant growth and development, starch synthesis, sugar transportation and accumulation of storage proteins which are essential for seed and pod development. On a combinatorial basis, our study provides a lncRNA and miRNA based regulatory insight into the genes governing seed and pod development in Pigeonpea.

Polynomial bases: Positivity and Schur multiplication

We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters, and Demazure atoms; the quasi-key, fundamental, and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and that an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason, and van Willigenburg’s (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.

On the Combinatorics of Gentle Algebras

AbstractFor $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

Analysis of students’ higher order thinking skills in solving basic combinatorics problems

Penelitian ini merupakan penelitian kualitatif berjenis studi kasus yang bertujuan untuk mendeskripsikan kemampuan berpikir tingkat tinggi mahasiswa dalam memecahkan masalah kombinatorika dasar. Subjek penelitiannya yaitu tiga mahasiswa program studi pendidikan matematika yang terdiri dari satu mahasiswa berkemampuan tinggi, satu mahasiswa berkemampuan sedang, dan satu mahasiswa berkemampuan rendah yang dipilih secara purposive. Penelitian dilakukan dengan cara menganalisis kemampuan berpikir kritis dan kreatif subjek penelitian dalam memecahkan masalah kaidah penghitungan, permutasi, dan kombinasi. Dari sisi kemampuan berpikir kritis, hasil penelitian menunjukkan bahwa kemampuan regulasi diri para mahasiswa masih rendah. Kemampuan berpikir kritis pada umumnya masih terbatas pada kemampuan menginterpretasi dan menganalisis masalah; hanya mahasiswa berkemampuan tinggi yang menunjukkan kemampuan dalam mengevaluasi masalah. Namun, rendahnya kemampuan regulasi diri menyebabkan ketiga mahasiswa tersebut belum mampu menyadari kesalahan pada strategi yang dibuat. Sedangkan, dari sisi kemampuan berpikir kreatif, para mahasiswa, terutama mahasiswa berkemampuan tinggi, mampu menghasilkan ide orisinal dalam memecahkan masalah kombinatorika dasar. Namun, rendahnya kemampuan berpikir fleksibel yang didukung oleh rendahnya kemampuan meregulasi diri menyebabkan ketiga mahasiswa terebut mengalami kesulitan dalam menghasilkan ide-ide yang berbeda.

Edge topology construction of Voronoi diagrams of spheres in non-general position

Although 3D Voronoi diagrams and medial axis transforms have numerous applications in biology, robotics, and manufacturing, most researchers use Voronoi diagrams of points instead of the true 3D input geometry, due to issues of robustness and scalability. In this paper, we present a robust sample-based GPU algorithm for calculating the full topology of Voronoi diagrams of non-general position spheres. Prior work demonstrated that the presence, geometry, and combinatorial basis of spheres that contribute to Voronoi vertices can be efficiently computed by shooting rays from each input sphere, mapping ray intersections with the nearest bisector surface to parametric bounding cubes, and analyzing the results. In this paper, we propose an algorithm on this parametric bounding cube to compute Voronoi edges in addition to the vertices. We successfully extract the full topology of the Voronoi diagram, including special cases such as isolated Voronoi edges that do not contain Voronoi vertices, more than three Voronoi edges emanating from a Voronoi vertex, and Voronoi edges that are shared by more than three Voronoi cells. Our GPU implementation efficiently and robustly handles all input, whether in general or non-general position, and finds all Voronoi vertices and edges, modulo the sampling density, including isolated disconnected edges.

Higher level vertex operators for $$U_q \left( \widehat{\mathfrak {sl}}_2\right) $$ U q sl ^ 2

We study graded nonlocal $\underline{\mathsf{q}}$-vertex algebras and we prove that they can be generated by certain sets of vertex operators. As an application, we consider the family of graded nonlocal $\underline{\mathsf{q}}$-vertex algebras $V_{c,1}$, $c\geq 1$, associated with the principal subspaces $W(c\Lambda_0)$ of the integrable highest weight $U_q (\hat{\mathfrak{sl}}_2)$-modules $L(c\Lambda_0)$. Using quantum integrability, we derive combinatorial bases for $V_{c,1}$ and compute the corresponding character formulae.

Multiple-solution problems in a statistics classroom: an example

ABSTRACTThe mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students’ understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact probability mass distribution for the sum of face values. Four different ways of solving the problem are discussed. The solutions span various basic concepts in different mathematical disciplines (sample space in probability theory, the probability generating function in statistics, integer partition in basic combinatorics and individual risk model in actuarial science) and thus promotes upper undergraduate students’ awareness of knowledge connections between their courses. All solutions of the example are implemented using the R statistical software package.

On principal realization of modules for the affine Lie algebra 𝐴₁⁽¹⁾ at the critical level

We present complete realization of irreducible A 1 ( 1 ) A_1^{(1)} –modules at the critical level in the principal gradation. Our construction uses vertex algebraic techniques, the theory of twisted modules and representations of Lie conformal superalgebras. We also provide an alternative Z-algebra approach to this construction. All irreducible highest weight A 1 ( 1 ) A_1^{(1)} –modules at the critical level are realized on the vector space M 1 2 + Z ( 1 ) ⊗ 2 M_{\tfrac {1}{2}+\mathbb {Z}}(1)^{\otimes 2} where M 1 2 + Z ( 1 ) M_{\tfrac {1}{2} + \mathbb {Z}}(1) is the polynomial ring C [ α ( − 1 / 2 ) , α ( − 3 / 2 ) , … ] {\mathbb {C}}[\alpha (-1/2), \alpha (-3/2),\dots ] . Explicit combinatorial bases for these modules are also given.

Bases of Feigin–Stoyanovsky’s type subspaces for $$C_\ell ^{(1)}$$ C ℓ ( 1 )

In this paper, we construct combinatorial bases of Feigin–Stoyanovsky’s type subspaces of standard modules for level k affine Lie algebra $$C_\ell ^{(1)}$$ . We prove spanning by using annihilating field $$x_\theta (z)^{k+1}$$ of standard modules. In the proof of linear independence, we use simple currents and intertwining operators whose existence is given by fusion rules.

Combinatorial bases of basic modules for affine Lie algebras Cn(1)

Lepowsky and Wilson initiated the approach to combinatorial Rogers-Ramanujan type identities via vertex operator constructions of standard (i.e., integrable highest weight) representations of affine Kac-Moody Lie algebras. Meurman and Primc developed further this approach for sl(2,C) ˜ by using vertex operator algebras and Verma modules. In this paper, we use the same method to construct combinatorial bases of basic modules for affine Lie algebras of type Cn(1) and, as a consequence, we obtain a series of Rogers-Ramanujan type identities. A major new insight is a combinatorial parametrization of leading terms of defining relations for level one standard modules for affine Lie algebra of type Cn(1).