arXiv subjects

Elijah Bodish

Publications and source records attributed to Elijah Bodish.

14 recordsLinked to original sources

arXiv2026

A Kohno--Drinfeld Theorem for iquantum Weyl groups

We prove that two finite-dimensional linear representations of the braid group are isomorphic. One representation comes from the monodromy of the boundary Casimir connection, and the other comes from the $\iota$quantum Weyl group. Both representations are defined for any split symmetric pair $\mathfrak{k}\subset \mathfrak{g}$ and for any integrable representation of $\mathfrak{k}$. Our proof uses $(O_m,\mathfrak{so}_{2n})$ spin Howe duality, so it is only for the pair $\mathfrak{so}_m\subset \mathfrak{sl}_m$.

math.QA↗

arXiv2026

Type $B$ Webs

We solve the type $B$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras." That is, we define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{so}_{2n+1})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{so}_{2n+1})$ tensor-generated by the fundamental representations. A consequence of our main result is an explicit construction of braid group symmetries for nonclassical finite-dimensional representations of the $\iota$quantum group ${U}^{\iota}_{-q^2}(\mathfrak{so}_m)$, which may be of independent interest.

math.RT↗

arXiv2025

MOY calculus in type D

We define a positive state sum for webs "of type D". These webs are graphs which mimic morphisms in the category of finite-dimensional quantum so(2N)-modules. From the state sum, we derive an invariant of framed unoriented links. After giving explicit details about some intertwiners in the category of quantum so(2N)-modules, we relate our state-sum link invariant with Reshetikhin--Turaev's invariant associated with quantum so(2N).

math.QA↗

arXiv2024

Cyclotomic nil-Brauer and Singular Soergel bimodules of type D

We introduce a new family of monoidal categories which are cyclotomic quotients of the nil-Brauer category. We construct a monoidal functor from the cyclotomic nil-Brauer category to another monoidal category constructed from singular Soergel bimodules of type D. We conjecture that our functor is an equivalence of categories. Although we can prove neither fullness nor faithfulness at this point, we are able to show that the functor induces an isomorphism at the level of Grothendieck rings. We compute these rings and their canonical bases, and give diagrammatic descriptions of the corresponding primitive idempotents.

math.RT↗

arXiv2024

Spin Link Homology

We put a new spin on Khovanov--Rozansky homology. That is, we equip $\Lambda^n$-colored $\mathfrak{sl}_{2n}$ Khovanov--Rozansky homology with an involution whose $\pm 1$-eigenspaces are link invariants. When $n=1,2,3$ (and assuming technical conjectures for $n \geq 4$), we prove that this refined invariant categorifies the spin-colored $\mathfrak{so}_{2n+1}$ quantum link polynomial. Along the way, we partially develop the theory of quantum $\mathfrak{so}_{2n+1}$ webs and make contact with $\iota$quantum groups.

math.QA↗

arXiv2024

Orthogonal Howe duality and dynamical (split) symmetric pairs

Inspired by Etingof--Varchenko's dynamical fusion, dynamical $R$-matrix, and dynamical Weyl group for Lie algebras, we introduce, for split symmetric pairs, versions of dynamical fusion, dynamical $K$-matrix, and dynamical Weyl group. We then turn to the study of $(\mathfrak{so}_{2n},O_m)$-duality and prove that the standard Knizhnik-Zamolodchikov and dynamical operators (both differential and difference) on the $\mathfrak{so}_{2n}$-side are exchanged with the symmetric pair analogs, for $O_m\subset GL_m$, on the $O_m$-side.

math.RT↗

arXiv2024

Orthogonal webs and semisimplification

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this category.

math.RT↗

arXiv2023

Webs for the Quantum Orthogonal Group

We give a generators and relations presentation for the full monoidal subcategory of representations of the quantum orthogonal group generated by the quantum exterior powers of the defining representation.

math.RT↗

arXiv2023

On a symplectic quantum Howe duality

We prove a nonsemisimple quantum version of Howe's duality with the rank 2n symplectic and the rank 2 special linear group acting on the exterior algebra of type C. We also discuss the first steps towards the symplectic analog of harmonic analysis on quantum spheres, give character formulas for various fundamental modules, and construct canonical bases of the exterior algebra.

math.RT↗

arXiv2022

Triple clasp formulas for $C_2$ webs

Using the light ladder basis for Kuperberg's $C_2$ webs, we derive triple clasp formulas for idempotents projecting to the top summand in each tensor product of fundamental representations. We then find explicit formulas for the coefficients occurring in the clasps, by computing these coefficients as local intersection forms. Our formulas provide further evidence for Elias's clasp conjecture, which was given for type $A$ webs, and suggests how to generalize the conjecture to non-simply laced types.

math.RT↗

arXiv2021

Triple Clasp Formulas for $G_2$

We use Kuperberg's diagrammatic description of the space of homomorphisms between fundamental representations of $G_2$ to give explicit recursive formulas for the idempotent projecting to the highest weight irreducible summand in each tensor product of fundamental representations.

math.RT↗

arXiv2021

Decreasing subsequences and Viennot for oscillating tableaux

We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points.

math.CO↗

arXiv2021

Web Calculus and Tilting Modules in Type $C_2$

Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $\mathfrak{so}_5\cong \mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_q\ne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $\mathcal{U}_q^{\mathbb{Z}}(\mathfrak{sp}_4)$.

math.RT↗

arXiv2021

Type $C$ Webs

We define a $\mathbb{C}(q)$-linear pivotal category $\mathbf{Web}(\mathfrak{sp}_{2n})$ and prove that it is equivalent to the full subcategory of finite-dimensional representations of $U_q(\mathfrak{sp}_{2n})$ tensor-generated by the fundamental representations. This answers the type $C$ case of the main open problem from Kuperberg's 1996 paper "Spiders for rank 2 Lie algebras" (arXiv:q-alg/9712003).

math.RT↗