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Journal articles on the topic 'Combinatorial representation theory'

Author: Grafiati

Published: 25 May 2024

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1

Ibarra, Andoni, and Thomas Mormann. "Una teoría combinatoria de las representaciones científicas." Crítica (México D. F. En línea) 32, no. 95 (January 7, 2000): 3–46. http://dx.doi.org/10.22201/iifs.18704905e.2000.874.

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The aim of this paper is to introduce a new concept of scientific representation into philosophy of science. The new concept -to be called homological or functorial representation- is a genuine generalization of the received notion of representation as a structure preserving map as it is used, for example, in the representational theory of measurement. It may be traced back, at least implicitly, to the works of Hertz and Duhem. A modern elaboration may be found in the foundational discipline of mathematical category theory. In contrast to the familiar concepts of representations, functorial representations do not depend on any notion of similarity, neither structural nor objectual one. Rather, functorial representation establish correlations between the structures of the representing and the represented domains. Thus, they may be said to form a class of quite "non-isomorphic" representations. Nevertheless, and this is the central claim of this paper, they are the most common type of representations used in science. In our paper we give some examples from mathematics and empirical science. One of the most interesting features of the new concept is that it leads in a natural way to a combinatorial theory of scientific representations, i.e. homological or functorial representations do not live in insulation, rather, they may be combined and connected in various ways thereby forming a net of interrelated representations. One of the most important tasks of a theory of scientific representations is to describe this realm of combinatorial possibilities in detail. Some first tentative steps towards this endeavour are done in our paper.

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2

Malinin, Dmitry. "One combinatorial construction in representation theory." European Journal of Combinatorics 80 (August 2019): 287–95. http://dx.doi.org/10.1016/j.ejc.2018.02.007.

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3

HSIEH, CHUN-CHUNG. "FIRST NON-VANISHING SELF-LINKING OF KNOTS (I) COMBINATORIC AND DIAGRAMMATIC STUDY." Journal of Knot Theory and Its Ramifications 20, no. 12 (December 2011): 1637–48. http://dx.doi.org/10.1142/s0218216511009510.

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In this paper, following the scheme of [Borromean rings and linkings, J. Geom. Phys.60 (2010) 823–831; Combinatoric and diagrammatic study in knot theory, J. Knot Theory Ramifications16 (2007) 1235–1253; Massey–Milnor linking = Chern–Simons–Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903], we study the first non-vanishing self-linkings of knots, aiming at the study of combinatorial formulae and diagrammatic representation. The upshot of perturbative quantum field theory is to compute the Feynman diagrams explicitly, though it is impossible in general. Along this line in this paper we could not only compute some Feynman diagrams, but also give the explicit and combinatorial formulae.

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4

Bremner, Murray R., Mikelis G. Bickis, and Mohsen Soltanifar. "Cayley’s hyperdeterminant: A combinatorial approach via representation theory." Linear Algebra and its Applications 437, no. 1 (July 2012): 94–112. http://dx.doi.org/10.1016/j.laa.2012.01.037.

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5

SHAI, OFFER. "The multidisciplinary combinatorial approach (MCA) and its applications in engineering." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 15, no. 2 (April 2001): 109–44. http://dx.doi.org/10.1017/s0890060401152030.

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The current paper describes the Multidisciplinary Combinatorial Approach (MCA), the idea of which is to develop discrete mathematical representations, called “Combinatorial Representations” (CR) and to represent with them various engineering systems. During the research, the properties and methods embedded in each representation and the connections between them were investigated thoroughly, after which they were associated with various engineering systems to solve related engineering problems. The CR developed up until now are based on graph theory, matroid theory, and discrete linear programming, whereas the current paper employs only the first two. The approach opens up new ways of working with representations, reasoning and design, some of which are reported in the paper, as follows: 1) Integrated multidisciplinary representation—systems which contain interrelating elements from different disciplines are represented by the same CR. Consequently, a uniform analysis process is performed on the representation, and thus on the whole system, irrespective of the specific disciplines, to which the elements belong. 2) Deriving known methods and theorems—new proofs to known methods and theorems are derived in a new way, this time on the basis of the combinatorial theorems embedded in the CR. This enables development of a meta-representation for engineering as a whole, through which the engineering reasoning becomes convenient. In the current paper, this issue is illustrated on structural analysis. 3) Deriving novel connections between remote fields—new connections are derived on the basis of the relations between the different combinatorial representations. An innovative connection between mechanisms and trusses, shown in the paper, has been derived on the basis of the mutual dualism between their corresponding CR. This new connection alone has opened several new avenues of research, since knowledge and algorithms from machine theory are now available for use in structural analysis and vice versa. Furthermore, it has opened opportunities for developing new design methods, in which, for instance, structures with special properties are developed on the basis of known mechanisms with special properties, as demonstrated in this paper. Conversely, one can use these techniques to develop special mechanisms from known trusses.

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Armenta, Marco, and Pierre-Marc Jodoin. "The Representation Theory of Neural Networks." Mathematics 9, no. 24 (December 13, 2021): 3216. http://dx.doi.org/10.3390/math9243216.

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In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.

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7

Griffeth, Stephen. "Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)." Proceedings of the Edinburgh Mathematical Society 53, no. 2 (April 30, 2010): 419–45. http://dx.doi.org/10.1017/s0013091508000904.

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AbstractThis paper aims to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category $\mathcal{O}$ for the rational Cherednik algebra of type G(r, p, n). As a first application, a self-contained and elementary proof of the analogue for the groups G(r, p, n), with r > 1, of Gordon's Theorem (previously Haiman's Conjecture) on the diagonal co-invariant ring is given. No restriction is imposed on p; the result for p ≠ r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's Conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.

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8

Capparelli, Stefano, Arne Meurman, Andrej Primc, and Mirko Primc. "New partition identities from \(C^{(1)}_\ell\)-modules." Glasnik Matematicki 57, no. 2 (December 30, 2022): 161–84. http://dx.doi.org/10.3336/gm.57.2.01.

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In this paper we conjecture combinatorial Rogers-Ramanujan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

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9

Vershik, A. M., and N. V. Tsilevich. "On the relationship between combinatorial functions and representation theory." Functional Analysis and Its Applications 51, no. 1 (January 2017): 22–31. http://dx.doi.org/10.1007/s10688-017-0165-4.

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10

Proctor, Robert A. "A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group." Canadian Journal of Mathematics 42, no. 1 (February 1, 1990): 28–49. http://dx.doi.org/10.4153/cjm-1990-002-1.

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This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

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11

Iyama, Osamu, Nathan Reading, Idun Reiten, and Hugh Thomas. "Lattice structure of Weyl groups via representation theory of preprojective algebras." Compositio Mathematica 154, no. 6 (May 16, 2018): 1269–305. http://dx.doi.org/10.1112/s0010437x18007078.

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This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

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12

Flath, Daniel, and Jacob Towber. "A Combinatorial Problem in the Representation Theory of SL(n)." Annals of Combinatorics 4, no. 3 (December 2000): 257–68. http://dx.doi.org/10.1007/pl00001280.

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13

Blanchard, J. L. "A Representation of Large Integers from Combinatorial Sieves." Journal of Number Theory 54, no. 2 (October 1995): 287–96. http://dx.doi.org/10.1006/jnth.1995.1119.

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14

MIEMIETZ, VANESSA, and WILL TURNER. "RATIONAL REPRESENTATIONS OF GL2." Glasgow Mathematical Journal 53, no. 2 (December 8, 2010): 257–75. http://dx.doi.org/10.1017/s0017089510000686.

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AbstractLet F be an algebraically closed field of characteristic p. We fashion an infinite dimensional basic algebra ←p(F), with a transparent combinatorial structure, which controls the rational representation theory of GL2(F).

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15

Moraglio, A., J. Togelius, and S. Silva. "Geometric Differential Evolution for Combinatorial and Programs Spaces." Evolutionary Computation 21, no. 4 (November 2013): 591–624. http://dx.doi.org/10.1162/evco_a_00099.

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Geometric differential evolution (GDE) is a recently introduced formal generalization of traditional differential evolution (DE) that can be used to derive specific differential evolution algorithms for both continuous and combinatorial spaces retaining the same geometric interpretation of the dynamics of the DE search across representations. In this article, we first review the theory behind the GDE algorithm, then, we use this framework to formally derive specific GDE for search spaces associated with binary strings, permutations, vectors of permutations and genetic programs. The resulting algorithms are representation-specific differential evolution algorithms searching the target spaces by acting directly on their underlying representations. We present experimental results for each of the new algorithms on a number of well-known problems comprising NK-landscapes, TSP, and Sudoku, for binary strings, permutations, and vectors of permutations. We also present results for the regression, artificial ant, parity, and multiplexer problems within the genetic programming domain. Experiments show that overall the new DE algorithms are competitive with well-tuned standard search algorithms.

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16

Li, Wen-Hui, Omran Kouba, Issam Kaddoura, and Feng Qi. "A further generalization of the Catalan numbers and its explicit formula and integral representation." Filomat 37, no. 19 (2023): 6505–24. http://dx.doi.org/10.2298/fil2319505l.

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In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy?s integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

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17

EL-GERESY, BAHER A., and ALIA I. ABDELMOTY. "TOWARDS A GENERAL THEORY FOR MODELLING QUALITATIVE SPACE." International Journal on Artificial Intelligence Tools 11, no. 03 (September 2002): 347–67. http://dx.doi.org/10.1142/s0218213002000939.

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Qualitative spatial representation and reasoning are techniques for modeling and manipulating objects and relationships in space. Finding ways for defining the complete and sound (physically plausible) set of relationships between spatial objects is a prerequisite for the development and realization of qualitative representation and reasoning formalisms. Establishing the set of sound relationships is a complicated task especially when complex objects are considered. Hence, current approaches to qualitative representation and reasoning are limited to handling simple spatial objects. In this paper, we introduce a constraint-based approach to qualitative representation of topological relationships by defining a set of general soundness rules. The rules reduce the combinatorial set of relations produced by the method to the complete and physically possible ones. The rules are general and apply to objects of arbitrary complexity and together with the representation and reasoning formalism form a theory for qualitative space.

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18

Blundell, Charles, Lars Buesing, Alex Davies, Petar Veličković, and Geordie Williamson. "Towards combinatorial invariance for Kazhdan-Lusztig polynomials." Representation Theory of the American Mathematical Society 26, no. 37 (November 16, 2022): 1145–91. http://dx.doi.org/10.1090/ert/624.

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Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.

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19

Batir, Necdet. "On some combinatorial identities and harmonic sums." International Journal of Number Theory 13, no. 07 (February 2017): 1695–709. http://dx.doi.org/10.1142/s179304211750097x.

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For any [Formula: see text] we first give new proofs for the following well-known combinatorial identities [Formula: see text] and [Formula: see text] and then we produce the generating function and an integral representation for [Formula: see text]. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that [Formula: see text] and [Formula: see text] where [Formula: see text] are generalized harmonic numbers defined below.

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20

Cao, Jian, Wen-Hui Li, Da-Wei Niu, Feng Qi, and Jiao-Lian Zhao. "A Brief Survey and an Analytic Generalization of the Catalan Numbers and Their Integral Representations." Mathematics 11, no. 8 (April 14, 2023): 1870. http://dx.doi.org/10.3390/math11081870.

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In the paper, the authors briefly survey several generalizations of the Catalan numbers in combinatorial number theory, analytically generalize the Catalan numbers, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy’s integral formula in the theory of complex functions, and point out potential directions for further study.

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21

Chevaleyre, Yann, Ulle Endriss, Jérôme Lang, and Nicolas Maudet. "Preference Handling in Combinatorial Domains: From AI to Social Choice." AI Magazine 29, no. 4 (December 28, 2008): 37. http://dx.doi.org/10.1609/aimag.v29i4.2201.

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In both individual and collective decision making, the space of alternatives from which the agent (or the group of agents) has to choose often has a combinatorial (or multi-attribute) structure. We give an introduction to preference handling in combinatorial domains in the context of collective decision making, and show that the considerable body of work on preference representation and elicitation that AI researchers have been working on for several years is particularly relevant. After giving an overview of languages for compact representation of preferences, we discuss problems in voting in combinatorial domains, and then focus on multiagent resource allocation and fair division. These issues belong to a larger field, known as computational social choice, that brings together ideas from AI and social choice theory, to investigate mechanisms for collective decision making from a computational point of view. We conclude by briefly describing some of the other research topics studied in computational social choice.

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22

Carvalho, Alda, Carlos Pereira dos Santos, Cátia Dias, Francisco Coelho, João Pedro Neto, Richard Nowakowski, and Sandra Vinagre. "On lattices from combinatorial game theory modularity and a representation theorem: Finite case." Theoretical Computer Science 527 (March 2014): 37–49. http://dx.doi.org/10.1016/j.tcs.2014.01.025.

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23

Çetinalp, Esra Kırmızı. "The n -Generalized Schützenberger-crossed product of monoids." Ukrains’kyi Matematychnyi Zhurnal 76, no. 2 (February 28, 2024): 276–88. http://dx.doi.org/10.3842/umzh.v76i2.7535.

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UDC 512.5 We study the n -generalized Schützenberger-crossed product from the viewpoint of combinatorial group theory and define a new version of this product. For given monoids of this new product, we obtain a representation of the n -generalized Schützenberger-crossed product of arbitrary monoids.

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24

Joh, Chang-Hyeon, Theo Arentze, and Harry Timmermans. "Modeling Individuals’ Activity-Travel Rescheduling Heuristics: Theory and Numerical Experiments." Transportation Research Record: Journal of the Transportation Research Board 1807, no. 1 (January 2002): 16–25. http://dx.doi.org/10.3141/1807-03.

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Previously, a theory of activity-travel rescheduling decisions was developed. This theory left open the problem of how individuals deal with the combinatorial problem of a very large solution space. Based on the argument that an appropriate algorithm should also be interpreted as a representation of an actual decision-making process, such an algorithm for activity-travel rescheduling is proposed here. Details are described, and a numerical illustration is provided to explore the face validity of the proposed algorithm.

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25

NATHANSON, MELVYN B. "PROBLEMS IN ADDITIVE NUMBER THEORY, IV: NETS IN GROUPS AND SHORTEST LENGTH g-ADIC REPRESENTATIONS." International Journal of Number Theory 07, no. 08 (December 2011): 1999–2017. http://dx.doi.org/10.1142/s1793042111004940.

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The number theoretic analog of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g ≥ 2, the study of h-nets in the additive group of integers with respect to the generating set Ag = {0} ∪ {± gi : i = 0, 1, 2, …} requires a knowledge of the word lengths of integers with respect to Ag. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.

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26

Dharwadker, Vinay. "Emotion in Motion: The Nāṭyashāstra, Darwin, and Affect Theory." Publications of the Modern Language Association of America 130, no. 5 (October 2015): 1381–404. http://dx.doi.org/10.1632/pmla.2015.130.5.1381.

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A work of classical Indian theory and practice, Bharata's Nāṭyashāstra offers a comprehensive account of emotion and of the production, communication, and reception of representations of it in dance, music, poetry, and theater. This essay examines remarkable points of convergence and divergence between the third-century Sanskrit text and three influential modern Euro-American accounts: Charles Darwin's mapping of involuntary expressions of emotion in human beings and animals, William James's aggregation of emotions in the stream of consciousness, and Sylvan Tomkins's atlas of primary affects that links neurobiology and cybernetics. My comparative analysis highlights the Nāṭyashāstra's contributions to our understanding of the connections of emotion to cognition, consciousness, and causality; of the combinatorial constitution of emotions; and of treatments of emotion in contemporary affect theory and performance theory. The essay concludes with an exploration of Bharata's and Aristotle's models of mimesis and of their mutual differences regarding the representation of emotion in the verbal and performing arts.

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Guillot, Pierre, and Ján Mináč. "Milnor K-theory and the graded representation ring." Journal of K-theory 13, no. 3 (May 12, 2014): 447–80. http://dx.doi.org/10.1017/is014004004jkt261.

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AbstractLet F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.

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GARVER, ALEXANDER, and THOMAS MCCONVILLE. "ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS." Glasgow Mathematical Journal 62, no. 1 (February 7, 2019): 147–82. http://dx.doi.org/10.1017/s0017089519000028.

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AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

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DUNIN-BARKOWSKI, P., A. SLEPTSOV, and A. SMIRNOV. "KONTSEVICH INTEGRAL FOR KNOTS AND VASSILIEV INVARIANTS." International Journal of Modern Physics A 28, no. 17 (July 10, 2013): 1330025. http://dx.doi.org/10.1142/s0217751x13300251.

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We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

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30

Naito, Satoshi. "A combinatorial identity for the derivative of a theta series of a finite type root lattice." Nagoya Mathematical Journal 172 (2003): 1–30. http://dx.doi.org/10.1017/s002776300000862x.

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AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.

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Precup, Martha, and Edward Richmond. "An equivariant basis for the cohomology of Springer fibers." Transactions of the American Mathematical Society, Series B 8, no. 17 (June 10, 2021): 481–509. http://dx.doi.org/10.1090/btran/57.

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Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for G L n ( C ) GL_n(\mathbb {C}) using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equivariant and ordinary cohomology rings of Springer fibers. As an application, we identify an explicit collection of (equivariant) Schubert classes whose images in the (equivariant) cohomology ring of a given Springer fiber form a basis.

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Severa, William, Ojas Parekh, Conrad D. James, and James B. Aimone. "A Combinatorial Model for Dentate Gyrus Sparse Coding." Neural Computation 29, no. 1 (January 2017): 94–117. http://dx.doi.org/10.1162/neco_a_00905.

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The dentate gyrus forms a critical link between the entorhinal cortex and CA3 by providing a sparse version of the signal. Concurrent with this increase in sparsity, a widely accepted theory suggests the dentate gyrus performs pattern separation—similar inputs yield decorrelated outputs. Although an active region of study and theory, few logically rigorous arguments detail the dentate gyrus’s (DG) coding. We suggest a theoretically tractable, combinatorial model for this action. The model provides formal methods for a highly redundant, arbitrarily sparse, and decorrelated output signal.To explore the value of this model framework, we assess how suitable it is for two notable aspects of DG coding: how it can handle the highly structured grid cell representation in the input entorhinal cortex region and the presence of adult neurogenesis, which has been proposed to produce a heterogeneous code in the DG. We find tailoring the model to grid cell input yields expansion parameters consistent with the literature. In addition, the heterogeneous coding reflects activity gradation observed experimentally. Finally, we connect this approach with more conventional binary threshold neural circuit models via a formal embedding.

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Brundan, Jonathan, Jonathan Comes, and Jonathan Robert Kujawa. "A Basis Theorem for the Degenerate Affine Oriented Brauer–Clifford Supercategory." Canadian Journal of Mathematics 71, no. 5 (March 7, 2019): 1061–101. http://dx.doi.org/10.4153/cjm-2018-030-8.

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AbstractWe introduce the oriented Brauer–Clifford and degenerate affine oriented Brauer–Clifford supercategories. These are diagrammatically defined monoidal supercategories that provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

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Keller, Bernhard, and Idun Reiten. "Acyclic Calabi–Yau categories." Compositio Mathematica 144, no. 5 (September 2008): 1332–48. http://dx.doi.org/10.1112/s0010437x08003540.

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AbstractWe prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if it contains a cluster-tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable category of maximal Cohen–Macaulay modules over a certain isolated singularity of dimension 3 is a cluster category. This implies the classification of the rigid Cohen–Macaulay modules first obtained by Iyama and Yoshino. As an application to the combinatorics of quiver mutation, we prove the non-acyclicity of the quivers of endomorphism algebras of cluster-tilting objects in the stable categories of representation-infinite preprojective algebras. No direct combinatorial proof is known as yet. In the appendix, Michel Van den Bergh gives an alternative proof of the main theorem by appealing to the universal property of the triangulated orbit category.

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Schwer, Petra. "Shadows in the Wild - Folded Galleries and Their Applications." Jahresbericht der Deutschen Mathematiker-Vereinigung 124, no. 1 (December 9, 2021): 3–41. http://dx.doi.org/10.1365/s13291-021-00244-2.

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AbstractThis survey is about combinatorial objects related to reflection groups and their applications in representation theory and arithmetic geometry. Coxeter groups and folded galleries in Coxeter complexes are introduced in detail and illustrated by examples. Further it is explained how they relate to retractions in Bruhat-Tits buildings and to the geometry of affine flag varieties and affine Grassmannians. The goal is to make these topics accessible to a wide audience.

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Mochizuki, Atsushi. "The Casson–Walker invariant of 3-manifolds with genus one open book decompositions." Journal of Knot Theory and Its Ramifications 28, no. 06 (May 2019): 1950018. http://dx.doi.org/10.1142/s0218216519500184.

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In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

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CASALI, MARIA RITA. "FROM FRAMED LINKS TO CRYSTALLIZATIONS OF BOUNDED 4-MANIFOLDS." Journal of Knot Theory and Its Ramifications 09, no. 04 (June 2000): 443–58. http://dx.doi.org/10.1142/s0218216500000220.

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It is well-known that every 3-manifold M3 may be represented by a framed link (L,c), which indicates the Dehn-surgery from [Formula: see text] to M3 = M3(L,c); moreover, M3 is the boundary of a PL 4-manifold M4 = M4(L, c), which is obtained from [Formula: see text] by adding 2-handles along the framed link (L, c). In this paper we study the relationships between the above representations and the representation theory of general PL-manifolds by edge-coloured graphs: in particular, we describe how to construct a 5-coloured graph representing M4=M4(L,c), directly from a planar diagram of (L,c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of M3=M3(L,c) and the regular genus of M4=M4(L,c) are obtained.

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KIRILLOV, ANATOL N. "DECOMPOSITION OF SYMMETRIC AND EXTERIOR POWERS OF THE ADJOINT REPRESENTATION OF ${\mathfrak g}l_N$ 1: UNIMODALITY OF PRINCIPAL SPECIALIZATION OF THE INTERNAL PRODUCT OF THE SCHUR FUNCTIONS." International Journal of Modern Physics A 07, supp01b (April 1992): 545–79. http://dx.doi.org/10.1142/s0217751x92003938.

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The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra [Formula: see text] into [Formula: see text]-irreducible components are considered. The exact formula for the principal specialization of the internal product of the Schur functions (similar to the formula for Kostka-Foulkes polynomials) is obtained by the purely combinatorial approach, based on the theory of rigged configurations. The stable behaviour of some polynomials is studied. Different examples are presented.

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ALEARDI, LUCA CASTELLI, OLIVIER DEVILLERS, and ABDELKRIM MEBARKI. "CATALOG-BASED REPRESENTATION OF 2D TRIANGULATIONS." International Journal of Computational Geometry & Applications 21, no. 04 (August 2011): 393–402. http://dx.doi.org/10.1142/s021819591100372x.

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Several Representations and Coding schemes have been proposed to represent efficiently 2D triangulations. In this paper, we propose a new practical approach to reduce the main memory space needed to represent an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometric information (vertex coordinates), since the combinatorial data represents the main part of the storage. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define and use stable catalogs of patches that are closed under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results that exhibit the practical gain of such methods.

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Satoh, Takao. "On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (May 2021): 338–63. http://dx.doi.org/10.1017/s0013091521000171.

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AbstractIn this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$ of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$ has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$.

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El-Shorbagy, M. A., and Aboul Ella Hassanien. "Particle Swarm Optimization from Theory to Applications." International Journal of Rough Sets and Data Analysis 5, no. 2 (April 2018): 1–24. http://dx.doi.org/10.4018/ijrsda.2018040101.

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Particle swarm optimization (PSO) is considered one of the most important methods in swarm intelligence. PSO is related to the study of swarms; where it is a simulation of bird flocks. It can be used to solve a wide variety of optimization problems such as unconstrained optimization problems, constrained optimization problems, nonlinear programming, multi-objective optimization, stochastic programming and combinatorial optimization problems. PSO has been presented in the literature and applied successfully in real life applications. In this paper, a comprehensive review of PSO as a well-known population-based optimization technique. The review starts by a brief introduction to the behavior of the PSO, then basic concepts and development of PSO are discussed, it's followed by the discussion of PSO inertia weight and constriction factor as well as issues related to parameter setting, selection and tuning, dynamic environments, and hybridization. Also, we introduced the other representation, convergence properties and the applications of PSO. Finally, conclusions and discussion are presented. Limitations to be addressed and the directions of research in the future are identified, and an extensive bibliography is also included.

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Cañadas, Agustín Moreno, and Odette M. Mendez. "Seaweeds Arising from Brauer Configuration Algebras." Mathematics 11, no. 8 (April 21, 2023): 1961. http://dx.doi.org/10.3390/math11081961.

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Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra Mat(n) introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one.

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Bessenrodt, Christine, Francesco Brenti, Alexander Kleshchev, and Arun Ram. "Combinatorial Representation Theory." Oberwolfach Reports, 2010, 799–882. http://dx.doi.org/10.4171/owr/2010/15.

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Grood, Cheryl. "A Specht Module Analog for the Rook Monoid." Electronic Journal of Combinatorics 9, no. 1 (December 18, 2001). http://dx.doi.org/10.37236/1619.

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The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

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"Representation theory: a combinatorial viewpoint." Choice Reviews Online 53, no. 05 (December 17, 2015): 53–2239. http://dx.doi.org/10.5860/choice.193758.

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Thiem, Nathaniel, and Vidya Venkateswaran. "Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices." Electronic Journal of Combinatorics 16, no. 1 (February 20, 2009). http://dx.doi.org/10.37236/112.

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It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

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Ly, Megan. "Shell Tableaux: A Set Partition Analog of Vacillating Tableaux." Electronic Journal of Combinatorics 26, no. 2 (May 3, 2019). http://dx.doi.org/10.37236/8241.

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Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph.

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48

Thiem, Nathaniel. "Branching rules in the ring of superclass functions of unipotent upper-triangular matrices." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AK,..., Proceedings (January 1, 2009). http://dx.doi.org/10.46298/dmtcs.2698.

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International audience It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.

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SAM, STEVEN V., and ANDREW SNOWDEN. "STABILITY PATTERNS IN REPRESENTATION THEORY." Forum of Mathematics, Sigma 3 (June 1, 2015). http://dx.doi.org/10.1017/fms.2015.10.

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We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

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Aliniaeifard, Farid, and Nathaniel Thiem. "Hopf Structures in the Representation Theory of Direct Products." Electronic Journal of Combinatorics 29, no. 4 (December 2, 2022). http://dx.doi.org/10.37236/11259.

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Combinatorial Hopf algebras give a linear algebraic structure to infinite families of combinatorial objects, a technique further enriched by the categorification of these structures via the representation theory of families of algebras. This paper examines a fundamental construction in group theory, the direct product, and how it can be used to build representation theoretic Hopf algebras out of towers of groups. A key special case gives us the noncommutative symmetric functions NSym, but there are many things that we can say for the general Hopf algebras, including the structure of their character groups and a formula for the antipode.

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