Duality in elliptic Ruijsenaars system and elliptic symmetric functions

The infinite number of particles limit in the dual to elliptic Ruijsenaars model (coordinate trigonometric degeneration of quantum double elliptic model) is proposed using the Nazarov-Sklyanin approach. For this purpose we describe double-elliptization of the Cherednik construction. Namely, we derive explicit expression in terms of the Cherednik operators, which reduces to the generating function of Dell commuting Hamiltonians on the space of symmetric functions. Although the double elliptic Cherednik operators do not commute, they can be used for construction of the N → ∞ limit.

Section: Discussionmentioning

confidence: 99%

“…up to some conjugation, which is explained in [38]. The statement is proven by the eigenvalue matching, however the explicit expression of the vertical generators in terms of the horizontal operators (elementary bosons p n , p ⊥ n ) in some nice form is still missing.…”

Section: Jhep12(2021)062mentioning

confidence: 99%

On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta

Grekov

Zotov

2021

“…• S R and their relatives are eigenfunctions of commuting set of Hamiltonians of quantum integrable many-body systems [27][28][29][30][31];…”

Section: Motivationmentioning

confidence: 99%

3-Schurs from explicit representation of Yangian $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$. Levels 1–5

Morozov,

Tselousov

2023

Self Cite

We suggest an ansatz for representation of affine Yangian $$ Y\left({\hat{\mathfrak{gl}}}_1\right) $$ Y gl ̂ 1 by differential operators in the triangular set of time-variables Pa,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting “cut-and-join” generators ψn of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods.

“…[11]). However the canonical coproduct structures chosen for affine Yangians [20] and quantum elliptic algebras [64][65][66] are more intricate.…”

Section: S(xy) = (−1) |X||y| S(y)s(x)mentioning

confidence: 99%

Toroidal and elliptic quiver BPS algebras and beyond

Galakhov

Yamazaki

2022

The quiver Yangian, an infinite-dimensional algebra introduced recently in [1], is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric quiver gauge theories, and their dimensionally-reduced counterparts. The analysis of supersymmetric gauge theories suggests that there exist even richer classes of algebras associated with higher-genus Riemann surfaces and generalized cohomology theories.