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In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog $\mathsf{T}(\widehat{\mathfrak{gl}}_{1|1})$ of Ding-Ioahara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed., Comment: 26 pages, 2 figures, v2: typos corrected, references added

We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual $p_k$ variables are accompanied by anti-commuting Grassmann variables $\theta_k$. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These new polynomials have similar properties to canonical Macdonald polynomials -- they respect two different orderings in the set of (super)-Young diagrams simultaneously.

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm's implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $\mathsf{Y}(\mathfrak{sl}_2)$, $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$ and $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$., Comment: 41 pages, 9 figures

BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters $\zeta$, jumping abruptly across the walls of marginal stability, which divide the $\zeta$-space into a collection of stability chambers -- ``phases'' of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the $\zeta$-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations -- starting with the example of affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the $\zeta$-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants., Comment: 36 pages, 7 figures, minor corrections, references added

BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. They arise in the limit $\hbar\longrightarrow 0$ from Macdonald polynomials, and generalize the well-known Jack polynomials ($\beta$-deformation of Schur functions), associated with $r=1$. For $r=2$ they approximate Macdonald polynomials with the accuracy $O(\hbar^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hbar$-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian $Y(\widehat{\mathfrak{gl}}_2)$, where $U^{(2)}$ are associated with the states $|\lambda\rangle$, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $p_{2n+1}$ can be expressed through mutually commuting operators from Yangian, however even time-variables $p_{2n}$ are inexpressible. Implications to higher $r$ become now straightforward, yet we describe them only in a sketchy way.

We review the main ideas underlying the emerging theory of Yangians -- the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems and the Duistermaat-Heckmann/equivariant integrals between the fixed points in the ADHM-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams) and Yangian generators describe ``instanton'' transitions between them. Detailed examples will be presented elsewhere.

We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,\theta_k)$. The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided., Comment: 27 pages, 3 figures

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank - which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit beta = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the R-matrix of U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). The resulting syste is the uplifting of the \widehat{\mathfrak{u}}_1 Wess-Zumino-Witten model. The solutions to the (q,t)-KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for 5d linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of KZE., Comment: 22 pages

R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A.~Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e.\ of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2,Z) group of DIM algebra automorphisms. We also construct the T-operators satisfying the RTT relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories., Comment: 31 pages