Last edited by Kihn
Tuesday, November 17, 2020 | History

8 edition of The q, t-Catalan numbers and the space of diagonal harmonics found in the catalog.

# The q, t-Catalan numbers and the space of diagonal harmonics

## by James Haglund

Written in English

Subjects:
• Symmetric functions,
• Combinatorial analysis,
• Polynomials,
• Harmonic functions

• Edition Notes

Includes bibliographical references (p. 163-167).

Classifications The Physical Object Statement James Haglund. Series University lecture series -- v. 41 LC Classifications QA212 .H34 2008 Pagination viii, 167 p. : Number of Pages 167 Open Library OL18504518M ISBN 10 0821844113 ISBN 10 9780821844113 LC Control Number 2007060570

The space of diagonal harmonics has emerged as one of the key ingredients in a program initiated by Garsia and Haiman to give a representation-theoretical proof of some conjectures in the theory of Macdonald polynomials. The study of this particular space has provided a remarkable display of connections between several areas. In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as. Course Control Number: Purpose: This is a graduate research seminar in which students will read and present papers from the literature. The theme is q -analogs in the theory of symmetric functions and the associated combinatorics, representation theory and geometry. Synonyms for diagonal in Free Thesaurus. Antonyms for diagonal. 26 synonyms for diagonal: slanting, angled, oblique, cross, crosswise, crossways, cater-cornered.

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### The q, t-Catalan numbers and the space of diagonal harmonics by James Haglund Download PDF EPUB FB2

The q, t-Catalan numbers and the space of diagonal harmonics: with an appendix on the combinatorics of Macdonald polynomials James Haglund This book contains detailed descriptions of the many exciting recent developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics.

The q, t-Catalan numbers and the space of diagonal harmonics; with an appendix on the combinatorics of Macdonald polynomials. Haglund, James. American Mathematical Society pages $Paperback University lecture series; v QA The q, t-Catalan numbers and the space of diagonal harmonics: with an appendix on the combinatorics of Macdonald polynomials / James Haglund. The (q;t)-Catalan Numbers and the Space of Diagonal Harmonics James Haglund University of Pennsylvania. Outline Intro to q-Analogues inv and maj q-Catalan Numbers MacMahon’s q-analogue TheCarlitz-Riordanq-analogue mension of the space of diagonal harmonics, as a vector space over Q. Keywords: q;t-Catalan numbers; sweep map; ~k-Dyck paths. Introduction In their study of the space DH n of diagonal harmonics [4], Garsia and Haiman introduced a q;t-analogue of the Catalan numbers, which they called the q;t-Catalan sequence. There are several equivalent characterizations of the (original) q;t-Catalan. In book: Recent Trends in Algebraic Combinatorics, pp The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square. Jim Haglund’ book: The q, t-Catalan Numbers and the Space of Diagonal Harmonics, with an Appendix on the Combinatorics of Macdonald Polynomials, University Lecture Series of the AMS, (with S. Hamel) Intersection of Modules Related to Macdonald’s Polynomials, Discrete. Hyperplane Arrangements and Diagonal Harmonics Drew Armstrongy Department of Mathematics, University of Miami, Coral Gables, FL, USA Abstract. InHaglund’s bounce statistic gave the ﬁrst combinatorial interpretation of the q;t-Catalan numbers and the Hilbert series of diagonal harmonics. We have that$\symS_n$act diagonally by permuting the variables in each alphabet, making the space of diagonal harmonics into a bigraded$\symS_n$-module$\mathcal{DH}_n.$One can show that$\mathcal{DH}_n$is also the ring of diagonal coinvariants. Destination page number Search scope Search Text Search scope Search Text. The$q,t$-Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Combinatorics of Macdonald Polynomials James Haglund Publication Year: This is a book on the combinatorics of the q,t-Catalan numbers and the space of diagonal harmonics. It is an expanded version of the lecture notes for a course on this topic given at the University of Pennsylvania in the spring of It includes an Appendix on some recent discoveries involving the combinatorics of Macdonald polynomials. Download full The Q T Catalan Numbers And The Space Of Diagonal Harmonics books PDF, EPUB, Tuebl, Textbook, Mobi or read online The Q T Catalan Numbers And The Space Of Diagonal Harmonics anytime and anywhere on any device. Get free access to the library by create an account, fast download and ads free. We cannot guarantee that every book is in. The Q,T-catalan numbers and the space of diagonal harmonics: with an appendix on the combinatorics of Macdonald polynomials This book contains detailed descriptions of the many exciting recent developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. The q,t-Catalan Numbers and the Space of Diagonal Harmonics in pdf format or ps format The official bound version of this book can be ordered from the AMS. It is part of the AMS University Lecture Series. This is a book on the combinatorics of the q, t-Catalan numbers and the space of diagonal harmonics. It is an expanded version of the lecture notes for a course on this topic given at the University of Pennsylvania in the spring of It includes an Appendix on some recent discoveries involving the combinatorics of Macdonald polynomials. Stanley's book is on reserve in the Physics/Math Library on the 3rd floor of DRL. Also, we will cover most of Chapter 1 from my book The q,t-Catalan Numbers and the Space of Diagonal Harmonics". In addition, some of the lectures will be taken from a pilot version. The operator ∇ of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some. DHA n is an S n-module called the space of diagonal harmonic alternants. We can define C n (q,t) to be the Hilbert series of this module relative to the double grading given by taking degrees in the x-variables and y-variables. Finally, we can use the area and bounce score statistics on Dyck paths to get a combinatorial definition of C n (q,t). Jim Haglund’ book: The q,t-Catalan Numbers and the Space of Diagonal Harmonics, with an Appendix on the Combinatorics of Macdonald Polynomials, University Lecture Series of the AMS, English Français (French). CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We introduce the q, t-Catalan numbers as the bivariate generating polynomials of two statistics on Dyck paths. We discuss some of their properties from a combinatorial point of view, e.g. a description by means of different statistics, a recursion, specialisations, and a possible generalisation to parking functions. Hot Topics Workshop on Algebraic Combinatorics The Combinatorics of q,t-Catalan Numbers and the Space of Diagonal Harmonics. / The Catalan numbers and Dyck paths; The q-Vandermonde convolution; Symmetric functions; The basics; Tableaux and Schur functions; Statistics on tableaux; Representation theory; The ring of coinvariants and the space of diagonal harmonics; The q, t-Catalan numbers; The bounce statistic; The special values t = 1 and t = 1/q; The symmetry problem. Topics: Kombinatorik, Graphentheorie, Algebra: Sonstiges, Kombinatorik / Catalan-Zahlen / Dyck-Pfade / Symmetrische Polynome, combinatorics / Catalan numbers /. The q,t-Catalan Numbers and the Space of Diagonal Harmonics - Jim Haglund Office hours: PM, Wednesdays, DRL 3C7 Homework: There will be a few homework assignments throughout the semester. q,t-Catalan numbers: A.M. Garsia and M. Haiman, A remarkable q,t -Catalan sequence and q -Lagrange inversion, J. Algebraic Comb. () J. Haglund, The q,t-Catalan Numbers and the Space of Diagonal Harmonics, AMS. pdf file). Sulzgruber, Robin () The symmetry of the q,t-Catalan numbers. Masterarbeit, Universität Wien. Fakultät für Mathematik BetreuerIn: Krattenthaler, Christian. If you have a disability and are having trouble accessing information on this website or need materials in an alternate format, contact [email protected] for [email protected] for assistance. One primary source for the lectures is my book The q,t-Catalan Numbers and the Space of Diagonal Harmonics. The official bound version of this book can now be ordered from the AMS. It is part of the AMS University Lecture Series. [15] Haglund, James The q, t-Catalan numbers and the space of diagonal harmonics, University Lecture Series, Vol American Mathematical Society, Providence, RI,viii+ pages (With an appendix on the combinatorics of Macdonald polynomials) | MR | Zbl The q, t-Catalan numbers arise naturally in the study of diagonal harmonics. They are conjectured to be the coefficient of the sign representation in the Frobenius series for the diagonal harmonics. The (q,t)-Catalan numbers are now known to be polynomials in q and t and there exists a proof in the article. The $$q$$, $$t$$-Catalan numbers and the space of diagonal harmonics. University Lecture Series, Vol. University Lecture Series, Vol. American Mathematical Society, Providence, RI. The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. Based on his study of the Hilbert scheme from algebraic geometry, Haiman [Invent. Math. (), pp. –] obtained a formula for the character of the space of diagonal harmonics under the diagonal action of the symmetric group, as a sum of Macdonald polynomials with rational coefficients. In this paper we show how Haiman's formula, combined with identities involving plethystic. Haglund “The q,t-Catalan Numbers and the Space of Diagonal Harmonics”, Chapter 1, Chapter 3, and pages of Appendix A (with Chapter 2 as background). • Module 3 (analytic): Pemantle-Wilson “Analytic combinatorics in Several Variables”, Chapters (with Chapter 2 as background). Macdonald E polynomials. The non-symmetric Macdonald polynomials were introduced in [], [] and [].The non-symmetric Macdonald polynomials are closely related with affine root systems, and the double affine Hecke definition is rather indirect, and does not give an efficient way of computing these non-symmetric polynomials. The q, t-Catalan numbers and the space of diagonal harmonics; with an appendix on the combinatorics of Macdonald polynomials. Positive polynomials and sums of squares. Mathematics for physical chemistry. A primer on linear models. Integrable systems and random matrices; proceedings. Additional results on diagonal harmonics. PostScript, PDF. t,q-Catalan numbers and the Hilbert scheme Discrete Math. (), Geometric interpretation of the t,q-Catalan number formula from A remarkable q,t-Catalan sequence and q-Lagrange inversion, below, using. If the coefficient of e n (X) is taken in this symmetric function, we have the definition of the q,t-Catalan numbers as they are defined in this web site. If the coefficient of e 1 n(X) is taken in this symmetric function, we have (conjectured) the Hilbert series for the space of diagonal harmonics and the dimension of this space is conjectured to be n-1, the number of rooted forests on n. TYPE-B Q,T-CATALAN NUMBERS AND BINOMIAL COEFFICIENTS Lingxi Lu A THESIS in character of the space of diagonal harmonics. De nition (Garsia & Haiman) 4 a l' l x a' Figure 2: Arm, leg, co-arm and co-leg. The q;t-Catalan sequence is de ned by setting C n(q. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math.$\mathbf. Based on the synthesis strategy of dual-harmonic phases discussed above, we will go a step further to investigate the wavefront manipulations of the two harmonics in the space domain.

Suppose that each meta-atom owns an independent time-varying reflection coefficient Γ p q t, where the superscript stands for its coordinates on the metasurface.Rational q;t-Catalan conjecture Conjecture (G., Negut,) The polynomial c m;n(q;t) can be computed as follows: c m;n(q;t) = (P m;n 1;e n); where P m;n is a certain operator acting on symmetric functions.

The right hand side is manifestly symmetric in q and t, and can be computed as an explicit sum of rational functions over standard Young.