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Algebraic Combinatorics and Finite Geometry
An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) Algebraic Combinatorics and Finite Geometry Leo Storme Ghent University Department of Mathematics: Analysis, Logic and Discrete Mathematics Krijgslaan 281 - Building S8 9000 Ghent Belgium Francqui Foundation, May 5, 2021 Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) ACKNOWLEDGEMENT Acknowledgement: A big thank you to Ferdinand Ihringer for allowing me to use drawings and latex code of his slide presentations of his lectures for Capita Selecta in Geometry (Ghent University). Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) OUTLINE 1 AN INTRODUCTION TO ALGEBRAIC GRAPH THEORY 2 ERDOS˝ -KO-RADO RESULTS 3 CAMERON-LIEBLER SETS IN PG(3; q) 4 CAMERON-LIEBLER k-SETS IN PG(n; q) Leo Storme Algebraic Combinatorics and Finite Geometry An Introduction to Algebraic Graph Theory Erdos-Ko-Rado˝ results Cameron-Liebler sets in PG(3; q) Cameron-Liebler k-sets in PG(n; q) DEFINITION A graph Γ = (X; ∼) consists of a set of vertices X and an anti-reflexive, symmetric adjacency relation ∼ ⊆ X × X. -
LINEAR ALGEBRA METHODS in COMBINATORICS László Babai
LINEAR ALGEBRA METHODS IN COMBINATORICS L´aszl´oBabai and P´eterFrankl Version 2.1∗ March 2020 ||||| ∗ Slight update of Version 2, 1992. ||||||||||||||||||||||| 1 c L´aszl´oBabai and P´eterFrankl. 1988, 1992, 2020. Preface Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent decades. The combinatorial nature of the determinant expansion (and the related difficulty in teaching it) may hint at the plausibility of some link between the two areas. A more profound connection, the use of determinants in combinatorial enumeration goes back at least to the work of Kirchhoff in the middle of the 19th century on counting spanning trees in an electrical network. It is much less known, however, that quite apart from the theory of determinants, the elements of the theory of linear spaces has found striking applications to the theory of families of finite sets. With a mere knowledge of the concept of linear independence, unexpected connections can be made between algebra and combinatorics, thus greatly enhancing the impact of each subject on the student's perception of beauty and sense of coherence in mathematics. If these adjectives seem inflated, the reader is kindly invited to open the first chapter of the book, read the first page to the point where the first result is stated (\No more than 32 clubs can be formed in Oddtown"), and try to prove it before reading on. (The effect would, of course, be magnified if the title of this volume did not give away where to look for clues.) What we have said so far may suggest that the best place to present this material is a mathematics enhancement program for motivated high school students. -
GL(N, Q)-Analogues of Factorization Problems in the Symmetric Group Joel Brewster Lewis, Alejandro H
GL(n, q)-analogues of factorization problems in the symmetric group Joel Brewster Lewis, Alejandro H. Morales To cite this version: Joel Brewster Lewis, Alejandro H. Morales. GL(n, q)-analogues of factorization problems in the sym- metric group. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. hal-02168127 HAL Id: hal-02168127 https://hal.archives-ouvertes.fr/hal-02168127 Submitted on 28 Jun 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2016 Vancouver, Canada DMTCS proc. BC, 2016, 755–766 GLn(Fq)-analogues of factorization problems in Sn Joel Brewster Lewis1y and Alejandro H. Morales2z 1 School of Mathematics, University of Minnesota, Twin Cities 2 Department of Mathematics, University of California, Los Angeles Abstract. We consider GLn(Fq)-analogues of certain factorization problems in the symmetric group Sn: rather than counting factorizations of the long cycle (1; 2; : : : ; n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. -
Newton Polytopes in Algebraic Combinatorics
Newton Polytopes in Algebraic Combinatorics Neriman Tokcan University of Michigan, Ann Arbor April 6, 2018 Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Outline Newton Polytopes Geometric Operations on Polytopes Saturated Newton Polytope (SNP) The ring of symmetric polynomials SNP of symmetric functions Partitions - Dominance Order Newton polytope of monomial symmetric function SNP and Schur Positivity Some other symmetric polynomials Additional results Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Newton Polytopes d A polytope is a subset of R ; d ≥ 1 that is a convex hull of finite d set of points in R : Convex polytopes are very useful for analyzing and solving polynomial equations. The interplay between polytopes and polynomials can be traced back to the work of Isaac Newton on plane curve singularities. P α The Newton polytope of a polynomial f = n c x α2Z ≥0 α 2 C[x1; :::; xn] is the convex hull of its exponent vectors, i.e., n Newton(f ) = conv(fα : cα 6= 0g) ⊆ R : Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Geometric operations on Polytopes d Let P and Q be two polytopes in R , then their Minkowski sum is given as P + Q = fp + q : p 2 P; q 2 Qg: The geometric operation of taking Minkowski sum of polytopes mirrors the algebraic operation of multiplying polynomials. Properties: Pn Pn n conv( i=1 Si ) = i=1 conv(Si ); for Si ⊆ R : Newton(fg) = Newton(f ) + Newton(g); for f ; g 2 C[x1; :::; xn]: Newton(f + g) = conv(Newton(f ) [ Newton(g)): Neriman Tokcan Newton Polytopes in Algebraic Combinatorics Saturated Newton Polytope{SNP A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. -
Algebraic Combinatorics
ALGEBRAIC COMBINATORICS c C D Go dsil To Gillian Preface There are p eople who feel that a combinatorial result should b e given a purely combinatorial pro of but I am not one of them For me the most interesting parts of combinatorics have always b een those overlapping other areas of mathematics This b o ok is an intro duction to some of the interac tions b etween algebra and combinatorics The rst half is devoted to the characteristic and matchings p olynomials of a graph and the second to p olynomial spaces However anyone who lo oks at the table of contents will realise that many other topics have found their way in and so I expand on this summary The characteristic p olynomial of a graph is the characteristic p olyno matrix The matchings p olynomial of a graph G with mial of its adjacency n vertices is b n2c X k n2k G k x p k =0 where pG k is the numb er of k matchings in G ie the numb er of sub graphs of G formed from k vertexdisjoint edges These denitions suggest that the characteristic p olynomial is an algebraic ob ject and the matchings p olynomial a combinatorial one Despite this these two p olynomials are closely related and therefore they have b een treated together In devel oping their theory we obtain as a bypro duct a numb er of results ab out orthogonal p olynomials The numb er of p erfect matchings in the comple ment of a graph can b e expressed as an integral involving the matchings by which we p olynomial This motivates the study of moment sequences mean sequences of combinatorial interest which can b e represented -
Arxiv:1407.5311V2 [Math.CO] 29 Apr 2017 SB-Labeling
SB-LABELINGS AND POSETS WITH EACH INTERVAL HOMOTOPY EQUIVALENT TO A SPHERE OR A BALL PATRICIA HERSH AND KAROLA MESZ´ AROS´ Abstract. We introduce a new class of edge labelings for locally finite lattices which we call SB-labelings. We prove for finite lattices which admit an SB-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices. Keywords: poset topology, M¨obiusfunction, crosscut complex, Tamari lattice, weak order 1. Introduction Anders Bj¨ornerand Curtis Greene have raised the following question (personal communi- cation of Bj¨orner;see also [15] by Greene). Question 1.1. Why are there so many posets with the property that every interval has M¨obiusfunction equaling 0; 1 or −1? Is there a unifying explanation? This paper introduces a new type of edge labeling that a finite lattice may have which we dub an SB-labeling. We prove for finite lattices admitting such a labeling that each open interval has order complex that is contractible or is homotopy equivalent to a sphere of some dimension. This immediately yields that the M¨obiusfunction only takes the values 0; ±1 on all intervals of the lattice. The construction and verification of validity of such labelings seems quite readily achievable on a variety of examples of interest. The name SB-labeling was chosen with S and B reflecting the possibility of spheres and balls, respectively. This method will easily yield that each interval in the weak Bruhat order of a finite Coxeter group, in the Tamari lattice, and in any finite distributive lattice is homotopy equivalent to a ball or a sphere of some dimension. -
Introduction to Algebraic Combinatorics: (Incomplete) Notes from a Course Taught by Jennifer Morse
INTRODUCTION TO ALGEBRAIC COMBINATORICS: (INCOMPLETE) NOTES FROM A COURSE TAUGHT BY JENNIFER MORSE GEORGE H. SEELINGER These are a set of incomplete notes from an introductory class on algebraic combinatorics I took with Dr. Jennifer Morse in Spring 2018. Especially early on in these notes, I have taken the liberty of skipping a lot of details, since I was mainly focused on understanding symmetric functions when writ- ing. Throughout I have assumed basic knowledge of the group theory of the symmetric group, ring theory of polynomial rings, and familiarity with set theoretic constructions, such as posets. A reader with a strong grasp on introductory enumerative combinatorics would probably have few problems skipping ahead to symmetric functions and referring back to the earlier sec- tions as necessary. I want to thank Matthew Lancellotti, Mojdeh Tarighat, and Per Alexan- dersson for helpful discussions, comments, and suggestions about these notes. Also, a special thank you to Jennifer Morse for teaching the class on which these notes are based and for many fruitful and enlightening conversations. In these notes, we use French notation for Ferrers diagrams and Young tableaux, so the Ferrers diagram of (5; 3; 3; 1) is We also frequently use one-line notation for permutations, so the permuta- tion σ = (4; 3; 5; 2; 1) 2 S5 has σ(1) = 4; σ(2) = 3; σ(3) = 5; σ(4) = 2; σ(5) = 1 0. Prelimaries This section is an introduction to some notions on permutations and par- titions. Most of the arguments are given in brief or not at all. A familiar reader can skip this section and refer back to it as necessary. -
Dynamical Algebraic Combinatorics and the Homomesy Phenomenon
Dynamical Algebraic Combinatorics and the Homomesy Phenomenon Tom Roby Abstract We survey recent work within the area of algebraic combinatorics that has the flavor of discrete dynamical systems, with a particular focus on the homomesy phenomenon codified in 2013 by James Propp and the author. Inthese situations, a group action on a set of combinatorial objects partitions them into orbits, and we search for statistics that are homomesic, i.e., have the same average value over each orbit. We give a number of examples, many very explicit, to illustrate the wide range of the phenomenon and its connections to other parts of combinatorics. In particular, we look at several actions that can be defined as a product of toggles, involutions on the set that make only local changes. This allows us to lift the well-known poset maps of rowmotion and promotion to the piecewise-linear and birational settings, where periodicity becomes much harder to prove, and homomesy continues to hold. Some of the examples have strong connections with the representation theory of semisimple Lie algebras, and others to cluster algebras via Y-systems. Key words: antichains, cluster algebras, homomesy, minuscule heaps, non-crossing partitions, orbits, order ideals, Panyushev complementation, permutations, poset, product of chains, promotion, root systems, rowmotion, Suter’s symmetry, toggle groups, tropicalization, Y-systems, Young’s Lattice, Young tableaux, Zamolodchikov periodicity. Mathematics Subject Classifications: 05E18, 06A11. 1 Introduction The term Dynamical Algebraic Combinatorics is meant to convey a range of phenomena involving actions on sets of discrete combinatorial objects, many of which can be built up by small local changes. -
Problems in Algebraic Combinatorics
PROBLEMS IN ALGEBRAIC COMBINATORICS C. D. Godsil 1 Combinatorics and Optimization University of Waterloo Waterloo, Ontario Canada N2L 3G1 [email protected] Submitted: July 10, 1994; Accepted: January 20, 1995. Abstract: This is a list of open problems, mainly in graph theory and all with an algebraic flavour. Except for 6.1, 7.1 and 12.2 they are either folklore, or are stolen from other people. AMS Classification Number: 05E99 1. Moore Graphs We define a Moore Graph to be a graph with diameter d and girth 2d +1. Somewhat surprisingly, any such graph must necessarily be regular (see [42]) and, given this, it is not hard to show that any Moore graph is distance regular. The complete graphs and odd cycles are trivial examples of Moore graphs. The Petersen and Hoffman-Singleton graphs are non-trivial examples. These examples were found by Hoffman and Singleton [23], where they showed that if X is a k-regular Moore graph with diameter two then k ∈{2, 3, 7, 57}. This immediately raises the following question: 1 Support from grant OGP0093041 of the National Sciences and Engineering Council of Canada is gratefully acknowledged. the electronic journal of combinatorics 2 (1995), #F1 2 1.1 Problem. Is there a regular graph with valency 57, diameter two and girth five? We summarise what is known. Bannai and Ito [4] and, independently, Damerell [12] showed that a Moore graph has diameter at most two. (For an exposition of this, see Chapter 23 in Biggs [7].) Aschbacher [2] proved that a Moore graph with valency 57 could not be distance transitive and G. -
Math 586 Algebraic Combinatorics Presentation Topics and Problems Instructor: Alexander Yong This Document Will Be Updated Throughout the Semester
Math 586 Algebraic combinatorics Presentation topics and problems Instructor: Alexander Yong This document will be updated throughout the semester. 1. Presentation topics Here are some possible topics. I am happy to make some suggestions about papers to look at to get you started. A presentation will be an explanation of the problem and main re- sults, as well as some computation of data. 1. Mobius inversion and the incidence algebra. 2. The combinatorial nullstellensatz (selected by Emily Heath). 3. The Jacobi-triple product identity 4. Geometric complexity theory. (Selected by William Linz) 5. Brill-Noether theory and set-valued tableaux. 6. Tropical Schur polynomials 7. K-orbit polynomials (e.g., involution Schubert polynomials) 8. Hessenberg varieties and Stanley's chromatic symmetric polynomial (Selected by Harshit Yadav) 9. CSM classes and the Aluffi-Mihalcea conjecture. 10. Monte Carlo methods and algebraic combinatorics. 11. Symmetric group characters as symmetric functions (Orellana-Zabrocki). Note, this topic is not asking about the textbook representation theory statement, but rather something novel due to the two listed mathematicians. (Selected by Simon Lin.) 12. Generalized permutahedra. (Selected by Nicole Yamzon.) 13. Matroid invariants. (Selected by Ramon Garcia Alvarez) 14. The amplituhedron (note the colloquium talk by Hugh Thomas) 15. Total positivity and the Grassmannian. (Selected by Gidon Orelowitz.) 16. \The Complexity of Generating Functions for Integer Points in Polyhedra and Beyond" by Alexander Barvinok (selected by Colleen Robichaux). 17. Applications of Lagrange inversion. 18. Asympototics of generating series. 19. Theory of species (generating series) 20. Tableau Kombinatorics (K-theory); we will partly cover this in class, but there's much more to talk about. -
18.212 S19 Algebraic Combinatorics, Lecture 39: Plane Partitions and More
18.212: Algebraic Combinatorics Andrew Lin Spring 2019 This class is being taught by Professor Postnikov. May 15, 2019 We’ll talk some more about plane partitions today! Remember that we take an m × n rectangle, where we have m columns and n rows. We fill the grid with weakly decreasing numbers from 0 to k, inclusive. Then the number of possible plane partitions, by the Lindstrom lemma, is the determinant of the k × k matrix C where entries are m + n c = : ij m + i − j MacMahon also gave us the formula m n k Y Y Y i + j + ` − 1 = : i + j + ` − 2 i=1 j=1 `=1 There are several geometric ways to think about this: plane partitions correspond to rhombus tilings of hexagons with side lengths m; n; k; m; n; k, as well as perfect matchings in a honeycomb graph. In fact, there’s another way to think about these plane partitions: we can also treat them as pseudo-line arrangements. Basically, pseudo-lines are like lines, but they don’t have to be straight. We can convert any rhombus tiling into a pseudo-line diagram! Here’s the process: start on one side of the region. For each pair of opposite sides of our hexagon, draw pseudo-lines by following the common edges of rhombi! For example, if we’re drawing pseudo-lines from top to bottom, start with a rhombus that has a horizontal edge along the top edge of the hexagon, and then connect this rhombus to the rhombus that shares the bottom horizontal edge. -
Recent Developments in Algebraic Combinatorics
Recent Developments in Algebraic Combinatorics Richard P. Stanley1 Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 e-mail: [email protected] version of 5 February 2004 Abstract We survey three recent developments in algebraic combinatorics. The first is the theory of cluster algebras and the Laurent phe- nomenon of Sergey Fomin and Andrei Zelevinsky. The second is the construction of toric Schur functions and their application to comput- ing three-point Gromov-Witten invariants, by Alexander Postnikov. The third development is the construction of intersection cohomol- ogy for nonrational fans by Paul Bressler and Valery Lunts and their application by Kalle Karu to the toric h-vector of a nonrational poly- tope. We also briefly discuss the “half hard Lefschetz theorem” of Ed Swartz and its application to matroid complexes. 1 Introduction. In a previous paper [32] we discussed three recent developments in algebraic combinatorics. In the present paper we consider three ad- ditional topics, namely, the Laurent phenomenon and its connection with Somos sequences and related sequences, the theory of toric Schur functions and its connection with the quantum cohomology of the Grassmannian and 3-point Gromov-Witten invariants, and the toric h-vector of a convex polytope. Note. The notation C, R, and Z, denotes the sets of complex numbers, real numbers, and integers, respectively. 1Partially supported by NSF grant #DMS-9988459. 1 2 The Laurent phenomenon. Consider the recurrence 2 n an−1an+1 = an +(−1) , n ≥ 1, (1) with the initial conditions a0 = 0, a1 = 1. A priori it isn’t evident that an is an integer for all n.