arXiv subjects

Tomasz Przezdziecki

Publications and source records attributed to Tomasz Przezdziecki.

11 recordsLinked to original sources

arXiv2026

Boundary $q$-characters of evaluation modules for split quantum affine symmetric pairs

We study evaluation modules for quantum symmetric pair coideal subalgebras of affine type $\mathsf{AI}$. By computing the action of the generators in Lu and Wang's Drinfeld-type presentation on Gelfand-Tsetlin bases, we determine the spectrum of a large commutative subalgebra arising from the Lu-Wang presentation. This leads to an explicit formula for boundary analogues of $q$-characters in the setting of quantum affine symmetric pairs. We interpret this formula combinatorially in terms of semistandard Young tableaux. Our results imply that boundary $q$-characters share familiar features with ordinary $q$-characters - such as a version of the highest weight property - yet they also display new phenomena, including an extra symmetry. In particular, we provide the first examples of boundary $q$-characters for quantum affine symmetric pairs that do not arise from restriction of ordinary $q$-characters, thereby revealing genuinely new structures in this new setting.

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arXiv2026

GKLO representations for shifted quantum affine symmetric pairs

In this note, we introduce shifted quantum affine symmetric pairs of split simply-laced type, and construct their GKLO representations, following similar recent developments in the case of shifted twisted Yangians. A full proof that our formulas yield a representation is given.

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arXiv2026

Compatibility of Drinfeld presentations and $q$-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a quasi-split affine quantum symmetric pair of type $\mathsf{AIII}$. This case is of particular interest thanks to the existence of geometric realizations and Schur--Weyl dualities. We establish factorization and coproduct formulae for the Drinfeld--Cartan series $\boldsymbolΘ_i(z)$ in the Lu--Pan--Wang--Zhang `new Drinfeld'-style presentation, generalizing the split type results from [Prz23, LP25a]. As an application, we construct a boundary analogue of the $q$-character map, and show that it is compatible with Frenkel and Reshetikhin's original $q$-character homomorphism.

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arXiv2025

GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian

We construct an analogue of Gerasimov-Kharchev-Lebedev-Oblezin (GKLO) representations for twisted Yangians of type $\mathsf{AI}$, using the recently found current presentation of these algebras due to Lu, Wang and Zhang. These new representations allow us to define interesting truncations of twisted Yangians, which, in the spirit of Ciccoli-Drinfeld-Gavarini quantum duality, reflect the Poisson geometry of homogeneous spaces. As our main result, we prove that a truncated twisted Yangian quantizes a scheme supported on quotients of transverse slices in the affine Grassmannian.

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arXiv2025

Drinfeld rational fractions for affine Kac-Moody quantum symmetric pairs

We formulate a precise connection between the new Drinfeld presentation of a quantum affine algebra $U_q\widehat{\mathfrak{g}}$ and the new Drinfeld presentation of affine coideal subalgebras of split type recently discovered by Lu and Wang. In particular, we establish a ``factorization formula'', expressing the commuting ``Drinfeld-Cartan''-type operators $Θ_{i,k}$ in a coideal subalgebra in terms of the corresponding Drinfeld generators of $U_q\widehat{\mathfrak{g}}$, modulo the ``Drinfeld positive half" of $U_q\widehat{\mathfrak{g}}$. We study the spectra of these operators on finite dimensional representations, and describe them in terms of rational functions with an extra symmetry. These results can be seen as the starting point of a $q$-character theory for affine Kac--Moody quantum symmetric pairs. Additionally, we prove a compatibility result linking Lusztig's and the Lu-Wang-Zhang braid group actions.

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arXiv2025

Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs

Let $(\mathbf{U}, \mathbf{U}^\imath)$ be a split affine quantum symmetric pair of type $\mathsf{B}_n^{(1)}, \mathsf{C}_n^{(1)}$ or $\mathsf{D}_n^{(1)}$. We prove factorization and coproduct formulae for the Drinfeld-Cartan operators $Θ_i(z)$ in the Lu-Wang Drinfeld-type presentation, generalizing the type $\mathsf{A}_n^{(1)}$ result from [Prz23]. As an application, we show that a boundary analogue of the $q$-character map, defined via the spectra of these operators, is compatible with the usual $q$-character map. As an auxiliary result, we also produce explicit reduced expressions for the fundamental weights in the extended affine Weyl groups of classical types.

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arXiv2023

Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra and $U_qL\mathfrak{g}$ the corresponding quantum affine algebra. We construct a functor ${}^θ{\sf F}$ between finite-dimensional modules over a quantum symmetric pair of affine type $U_q\mathfrak{k}\subset U_qL{\mathfrak{g}}$ and an orientifold KLR algebra arising from a framed quiver with a contravariant involution, providing a boundary analogue of Kang-Kashiwara-Kim-Oh generalized Schur-Weyl duality. With respect to their construction, our combinatorial model is further enriched with the poles of a trigonometric K-matrix intertwining the action of $U_q\mathfrak{k}$ on finite-dimensional $U_qL{\mathfrak{g}}$-modules. By construction, ${}^θ{\sf F}$ is naturally compatible with the Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of monoidal categories, ${}^θ{\sf F}$ is a functor of module categories. Relying on a suitable isomorphism à la Brundan-Kleshchev-Rouquier, we prove that ${}^θ{\sf F}$ recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in quasi-split type $\sf AIII$.

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arXiv2023

Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial.

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arXiv2020

The combinatorics of $\mathbb{C}^*$-fixed points in generalized Calogero-Moser spaces and Hilbert schemes

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the $q$-hook formula.

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arXiv2020

Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the Calogero-Moser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding.

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arXiv2019

Quiver Schur algebras and cohomological Hall algebras

We establish a connection between a generalization of KLR algebras, called quiver Schur algebras, and the cohomological Hall algebras of Kontsevich and Soibelman. More specifically, we realize quiver Schur algebras as algebras of multiplication and comultiplication operators on the CoHA, and reinterpret the shuffle description of the CoHA in terms of Demazure operators. We introduce ``mixed quiver Schur algebras" associated to quivers with a contravariant involution, and show that they are related, in an analogous way, to the cohomological Hall modules defined by Young. We also obtain a geometric realization of the modified quiver Schur algebra, which appeared in a version of the Brundan-Kleshchev-Rouquier isomorphism for the affine q-Schur algebra due to Miemietz and Stroppel.

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