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Abstract We construct a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the positive subalgebras of quantum loop algebras of type $$B_{n}$$ and $$G_{2}$$, as well as their Lusztig and RTT (for type $$B_{n}$$ only) integral forms, in the new Drinfeld realization. We also establish a shuffle algebra realization of these $${\mathbb {Q}}(v)$$-algebras (proved earlier in [26] by completely different tools) and generalize the latter to the above $${{\mathbb {Z}}}[v,v^{-1}]$$-forms. The rational counterparts provide shuffle algebra realizations of positive subalgebras of type $$B_{n}$$ and $$G_{2}$$ Yangians and their Drinfeld-Gavarini duals. All of this generalizes the type $$A_{n}$$ results of [30].
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Zonotopal Algebras, Orbit Harmonics, and Donaldson–Thomas Invariants of Symmetric Quivers
Abstract We apply the method of orbit harmonics to the set of break divisors and orientable divisors on graphs to obtain the central and external zonotopal algebras, respectively. We then relate a construction of Efimov in the context of cohomological Hall algebras to the central zonotopal algebra of a graph $$G_{Q,\gamma }$$ constructed from a symmetric quiver $$Q$$ with enough loops and a dimension vector $$\gamma $$. This provides a concrete combinatorial perspective on the former work, allowing us to identify the quantum Donaldson–Thomas (DT) invariants as the Hilbert series of the space of $$S_{\gamma }$$-invariants of the Postnikov–Shapiro slim subgraph space attached to $$G_{Q,\gamma }$$. The connection with orbit harmonics in turn allows us to give a manifestly nonnegative combinatorial interpretation to numerical DT invariants as the number of $$S_{\gamma }$$-orbits under the permutation action on the set of break divisors on $$G$$. We conclude with several representation-theoretic consequences, whose combinatorial ramifications may be of independent interest.
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- Award ID(s):
- 2246846
- PAR ID:
- 10402429
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 23
- ISSN:
- 1073-7928
- Format(s):
- Medium: X Size: p. 20169-20210
- Size(s):
- p. 20169-20210
- Sponsoring Org:
- National Science Foundation
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