Nabla Operator: $\nabla$

The $\nabla$ operator on symmetric function was first considered by me in 1994 (see this letter sent to A. Grasia), and then studied in collaboration with Adriano Garsia for a few years, before it got an official birth in a joint publication with Garsia: Science Fiction and Macdonald’s Polynomials, in Algebraic Methods and q-Special Functions, R.Floreanini, L.Vinet (eds.), CRM Proceedings & Lecture Notes, AMS, 1999, 1–52. It is defined in terms of the combinatorial Macdonald polynomials $H_\mu(\mathbf{x};q,t)$, by the fact that it affords these as joint eigenfunctions

$\nabla H_\mu(\mathbf{x};q,t) = q^{n(\mu’)}t^{n(\mu)} H_\mu(\mathbf{x};q,t)$

where $n(\mu):=\sum_i (i-1)\mu_i$, for all partition $\mu$ of $n$. Among the many roles it plays, there is the fact that $\nabla(e_n)$ gives the Frobenius transform of the bigraded character of the $\mathbb{S}_n$-module $\mathcal{D}_n$, of diagonal harmonic polynomials. This means that the coefficients of the Schur function expansion of $\nabla(e_n)$ correspond to graded multiplicities of irreducible representations of these modules. For example,

\begin{align*}\nabla(e_3(\mathbf{x})) = s_3(\mathbf{x})+ \left( {q}^{2}+qt+{t}^{2}+q+t \right) s_{{21}}(\mathbf{x})\\ \qquad\qquad\qquad + \left( {q}^{3}+{q}^{2}t+q{t}^{2}+{t}^{3}+qt \right) s_{{111}}(\mathbf{x})\end{align*}

In particular, the coefficient of $e_n(\mathbf{x})=s_{11\cdots1}(\mathbf{x})$ in $\nabla(e_3(\mathbf{x}))$ is the famous $q,t$-Catalan polynomial, in which the coefficient of $q^it^j$ is the dimension of the homogeneous component of bidegree $(i,j)$ of the alternating component of $\mathcal{D}_n$.  Many other important operators arise in a similar manner.

With Haiman,  Garsia, and  Tesler, we have interesting Schur-positivity conjectures about $\nabla(s_\mu)$, stated in  Identities and Positivity Conjectures for Some Remarkable Operators in the Theory of Symmetric Functions, Methods and Applications of Analysis, vol. 6, no. 3 (1999), 363–420.

Some papers where it plays a significant role

  • M. D’Adderio and M. Romero, New Identities for Theta Operators, Trans. Amer. Math. Soc. 376 (2023), no. 8, 5775–5807.
  • J. Blasiak, M. Haiman, J. Morse, A. Pun, and G.H. Seelinger,  A proof of the extended delta conjecture, Forum Math. Pi 11 (2023), Paper No. e6, 28 pp.
  • M. D’Adderio and A.. Mellit, A proof of the delta conjecture, Adv. Math. 402 (2022), Paper No. 108342, 17 pp.
  • A. Mellit, Toric braids and (m,n)–parking functions, Duke Math. J.170(2021), no.18, 4123–4169.
  • A. Mellit, Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures), Invent. math. (2020) 221:301–327.
  • A.T. Wilson, Torus link homology and the nabla operator, J. Combin. Theory Ser. A 154 (2018), 129–144.
  • Sh. Shakirov, Colored knot amplitudes and Hall-Littlewood polynomials, (arXiv:1308.3838) 2013, 20 pages.
  • Y. Kim, A Parking Function Setting for Nabla Images of Schur Functions, DMTCS proc. AS, 2013, 1035–1046
  • O. Blondeau-Fournier, L. Lapointe, and P. Mathieu, Double Macdonald polynomials as the stable limit of Macdonald superpolynomials, J. Algebraic Combin. 41 (2015), no. 2, 397–459.
  • K. Lee, L. Li and N. Loehr, Limits of Modified Higher $(q,t)$-Catalan Numbers, (arXiv:1110.5850), 2011.
  • James Haglund, Jennifer Morse and Mike Zabrocki, A compositional shuffle conjecture specifying touch points of the Dyck path, (arXiv:1008.0828), 2010, 20 pages
  • D. Armstrong, Hyperplane Arrangements and Diagonal Harmonics, (arXiv:1005.1949), 2010, 27 pages.
  • N. Bergeron, F. Descouens and M. Zabrocki, A generalization of $(q,t)$-Catalan and nabla operators, DMTCS proc. AJ, 2008, 513–528.
  • N. Loehr and G. Warrington, Nested Quantum Dyck Paths and  $\nabla(s_\lambda)$ International Mathematics Research Notices, Vol. 2008, Article ID rnm157, 29 pages. (arXiv:0705.4608)
  • M. Can and N. Loehr, A proof of the $q,t$-square conjecture, Journal of Combinatorial Theory, Series A, Vol. 113, Issue 7 (2006), 1419-1434.
  • N. Loehr, Combinatorics of $q,t$-parking functions, Advances in Applied Mathematics 34 (2005) 408–425.
  • J. Haglund, M. Haiman, N. Loehr, J. B. Remmel and A. Ulyanov, A Combinatorial Formula for the Character of the Diagonal Coinvariants, (arXiv:math/0310424), (2003), 31 pages.
  • C. Lenart, Lagrange Inversion and Schur Functions, Journal of Algebraic Combinatorics, Vol. 11, No 1 (2000).
  • (with A. M. Garsia, M. Haiman and G. Tesler)  Identities and Positivity Conjectures for Some Remarkable Operators in the Theory of Symmetric Functions,  Methods and Applications of Analysis, Volume 6 Number 3 (1999), 363–420.
  • M. Haiman, A. M. Garsia and G. Tesler, Explicit plethysic formulas for Macdonald $q,t$-Kostka coefficients, The Andrews Festschrift. Seminaire Lotharingien 42 (1999), electronic, 45pp.