The \(q,t\)-Catalan numbers and the space of diagonal harmonics. With an appendix on the combinatorics of Macdonald polynomials.

*(English)*Zbl 1142.05074
University Lecture Series 41. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4411-3/pbk). viii, 167 p. (2008).

In 1988 Macdonald introduced a family of multivariate orthogonal polynomials which became known as Macdonal polynomials. Those remarkable objects arise in several areas of mathematics, such as mathematical physics, harmonic analysis and geometry of Hilbert schemes. Other important families of polynomials, such as Jack polynomials and Hall-Littlewood polynomials are known to be specializations of Macdonald polynomials.

For a long time the combinatorial description of Macdonald polynomials remained a mystery, until in 2004 the author came up with such a description, which was later proved in joint work with M. Haiman and N. Loehr [J. Am. Math. Soc. 18, No. 3, 735–761 (2005; Zbl 1061.05101)] The proof and consequences of this formula are discussed in detail in Appendix A. The primary subject of the book, however, are diagonal harmonics, the study of which is intimately related with that of Macdonald polynomial, and in fact has inspired the above mentioned breakthrough of the author. The study was initiated by Garsia and Haiman and motivated a lot of recent research in algebraic combinatorics.

Consider the space \(\mathbb C[X_n, Y_n]\) of polynomials in two independent \(n\)-sets of variables. The symmetric group \(S_n\) acts on it diagonally, i.e. simultaneously permuting each set of variables. One can consider then the invariants \(\mathbb C[X_n, Y_n]^{S_n}\) of this action, which happen to be generated as a ring by polarized power sums \(p_{h,k} = \sum_{i=1}^n x_i^h y_i^k\). Two important objects to consider then are the ring of diagonal coinvariants

\[ DR_n = C[X_n, Y_n]/C[X_n, Y_n]^{S_n} \] and the space of diagonal harmonics

\[ DH_n = \biggl\{f \in C[X_n, Y_n] \;\biggl|\;\sum_{i=1}^n \frac{\partial^h}{x_i^h} \frac{\partial^k}{y_i^k} f = 0\biggr\}. \] Let \(DH_n^{\varepsilon}\) denote the subspace of anti-symmetric elements of \(DH_n\). One can use the bigrading of this space to introduce the \({q,t}\)-Catalan numbers as Hilbert series

\[ C_n(q,t) = H(DH_n^{\varepsilon};q,t). \] The name is justified by the fact that \(C_n = C_n(1,1)\) are the usual Catalan numbers as was shown by Garsia and Haiman. Study of \(q,t\)-Catalan numbers is one of the major objectives of the book. In particular a combinatorial description is obtained in terms of area and bounce statistics on Dyck paths. The description was conjectured by the author and proved in a joint work with A. M. Garsia [Discrete Math. 256, No. 3, 677–717 (2002; Zbl 1028.05115)]. While area is a natural statistic used before, bounce as well as another statistic called dinv are new. Further topics considered in the book are \(q,t\)-SchrĂ¶der numbers and relation between diagonal harmonics and parking functions.

Overall the book is an excellent introduction into the combinatorial side of a beautiful and very active area of research. Based on the author’s lecture notes for a course taught at the University of Pennsylvania in Spring 2004, this book can be used for self-study and as a reference for researchers in this area.

For a long time the combinatorial description of Macdonald polynomials remained a mystery, until in 2004 the author came up with such a description, which was later proved in joint work with M. Haiman and N. Loehr [J. Am. Math. Soc. 18, No. 3, 735–761 (2005; Zbl 1061.05101)] The proof and consequences of this formula are discussed in detail in Appendix A. The primary subject of the book, however, are diagonal harmonics, the study of which is intimately related with that of Macdonald polynomial, and in fact has inspired the above mentioned breakthrough of the author. The study was initiated by Garsia and Haiman and motivated a lot of recent research in algebraic combinatorics.

Consider the space \(\mathbb C[X_n, Y_n]\) of polynomials in two independent \(n\)-sets of variables. The symmetric group \(S_n\) acts on it diagonally, i.e. simultaneously permuting each set of variables. One can consider then the invariants \(\mathbb C[X_n, Y_n]^{S_n}\) of this action, which happen to be generated as a ring by polarized power sums \(p_{h,k} = \sum_{i=1}^n x_i^h y_i^k\). Two important objects to consider then are the ring of diagonal coinvariants

\[ DR_n = C[X_n, Y_n]/C[X_n, Y_n]^{S_n} \] and the space of diagonal harmonics

\[ DH_n = \biggl\{f \in C[X_n, Y_n] \;\biggl|\;\sum_{i=1}^n \frac{\partial^h}{x_i^h} \frac{\partial^k}{y_i^k} f = 0\biggr\}. \] Let \(DH_n^{\varepsilon}\) denote the subspace of anti-symmetric elements of \(DH_n\). One can use the bigrading of this space to introduce the \({q,t}\)-Catalan numbers as Hilbert series

\[ C_n(q,t) = H(DH_n^{\varepsilon};q,t). \] The name is justified by the fact that \(C_n = C_n(1,1)\) are the usual Catalan numbers as was shown by Garsia and Haiman. Study of \(q,t\)-Catalan numbers is one of the major objectives of the book. In particular a combinatorial description is obtained in terms of area and bounce statistics on Dyck paths. The description was conjectured by the author and proved in a joint work with A. M. Garsia [Discrete Math. 256, No. 3, 677–717 (2002; Zbl 1028.05115)]. While area is a natural statistic used before, bounce as well as another statistic called dinv are new. Further topics considered in the book are \(q,t\)-SchrĂ¶der numbers and relation between diagonal harmonics and parking functions.

Overall the book is an excellent introduction into the combinatorial side of a beautiful and very active area of research. Based on the author’s lecture notes for a course taught at the University of Pennsylvania in Spring 2004, this book can be used for self-study and as a reference for researchers in this area.

Reviewer: Pavlo Pylyavskyy (Ann Arbor)