SIGMA 22 (2026), 076, 50 pages      arXiv:2405.00756      https://doi.org/10.3842/SIGMA.2026.076

Murnaghan-Type Representations for the Positive Elliptic Hall Algebra

Milo Bechtloff Weising
Department of Mathematics, Virginia Tech, 460 McBryde Hall, 225 Stanger Street,Blacksburg, VA 24061-1026, USA

Received July 01, 2025, in final form July 27, 2026; Published online August 14, 2026

Abstract
We construct a new family of graded representations $\widetilde{W}_{\lambda}$ for the positive elliptic Hall algebra $\mathcal{E}^{+}$ indexed by Young diagrams $\lambda$ which generalize the standard $\mathcal{E}^{+}$ action on symmetric functions. These representations have homogeneous bases of eigenvectors for the action of the Macdonald element $P_{0,1} \in \mathcal{E}^{+}$ with distinct $\mathbb{Q}(q,t)$-rational spectrum generalizing the symmetric Macdonald functions. The analysis of the structure of these representations exhibits interesting combinatorics arising from the limits of periodic standard Young tableaux. We find an explicit combinatorial rule for the action of the multiplication operators $e_r^{\bullet}$ generalizing the Pieri rule for symmetric Macdonald functions.

Key words: elliptic Hall algebra; vector-valued; double affine Hecke algebra; Macdonald polynomials.

pdf (736 kb)   tex (50 kb)  

References

1. Bechtloff Weising M., Double dyck path algebra representations from DAHA, arXiv:2402.02843.
2. Bergeron F., Garsia A.M., Science fiction and Macdonald's polynomials, in Algebraic Methods and $q$-Special Functions (Montréal, QC, 1996), CRM Proc. Lecture Notes, Vol. 22, American Mathematical Society, Providence, RI, 1999, 1-52, arXiv:math.CO/9809128.
3. Bergeron F., Garsia A.M., Haiman M., Tesler G., Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999), 363-420.
4. Burban I., Schiffmann O., On the Hall algebra of an elliptic curve, I, Duke Math. J. 161 (2012), 1171-1231, arXiv:math.AG/0505148.
5. Carlsson E., Gorsky E., Mellit A., The $\mathbb{A}_{q,t}$ algebra and parabolic flag Hilbert schemes, Math. Ann. 376 (2020), 1303-1336, arXiv:1710.01407.
6. Carlsson E., Mellit A., A proof of the shuffle conjecture, J. Amer. Math. Soc. 31 (2018), 661-697, arXiv:1508.06239.
7. Cherednik I., Double affine Hecke algebras and difference Fourier transforms, Invent. Math. 152 (2003), 213-303, arXiv:math.QA/0110024.
8. Cherednik I., Double affine Hecke algebras, London Math. Soc. Lecture Note Ser., Vol. 319, Cambridge University Press, Cambridge, 2005.
9. Colmenarejo L., Dunkl C.F., Luque J.-G., Connections between vector-valued and highest weight Jack and Macdonald polynomials, Ann. Inst. Henri Poincaré D 9 (2022), 297-348, arXiv:1907.04631.
10. Dunkl C.F., A positive-definite inner product for vector-valued Macdonald polynomials, Sém. Lothar. Combin. 80 (2019), B80a, 26 pages, arXiv:1808.05251.
11. Dunkl C.F., Luque J.-G., Vector valued Macdonald polynomials, Sém. Lothar. Combin. 66 (2011), B66b, 68 pages, arXiv:1106.0875.
12. Feigin B., Feigin E., Jimbo M., Miwa T., Mukhin E., Quantum continuous $\mathfrak{gl}_\infty$: semiinfinite construction of representations, Kyoto J. Math. 51 (2011), 337-364, arXiv:1002.3100.
13. Feigin B., Jimbo M., Miwa T., Mukhin E., Quantum toroidal $\mathfrak{gl}_1$-algebra: plane partitions, Kyoto J. Math. 52 (2012), 621-659, arXiv:1110.5310.
14. Haglund J., Haiman M., Loehr N., A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math. 130 (2008), 359-383, arXiv:2008.0015.
15. Haiman M., Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006, arXiv:math.AG/0010246.
16. Macdonald I., A new class of symmetric functions, Sém. Lothar. Combin. 20 (1988), B20a, 41 pages, arXiv:2408.01704.
17. Murnaghan F.D., The analysis of the Kronecker product of irreducible representations of the symmetric group, Amer. J. Math. 60 (1938), 761-784.
18. Murphy G.E., On the representation theory of the symmetric groups and associated Hecke algebras, J. Algebra 152 (1992), 492-513.
19. Schiffmann O., Vasserot E., The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compos. Math. 147 (2011), 188-234, arXiv:0802.4001.
20. Schiffmann O., Vasserot E., The elliptic Hall algebra and the $K$-theory of the Hilbert scheme of ${\mathbb A}^2$, Duke Math. J. 162 (2013), 279-366, arXiv:0905.2555.
21. Stembridge J.R., A weighted enumeration of maximal chains in the Bruhat order, J. Algebraic Combin. 15 (2002), 291-301.