arXiv subjects

Alexander P. Ellis

Publications and source records attributed to Alexander P. Ellis.

9 recordsLinked to original sources

arXiv2026

SWE-Prot\'eg\'e: Learning to Selectively Collaborate With an Expert Unlocks Small Language Models as Software Engineering Agents

Small language models (SLMs) offer compelling advantages in cost, latency, and adaptability, but have so far lagged behind larger models on long-horizon software engineering tasks such as SWE-bench, where they suffer from pervasive action looping and low resolution rates. We introduce SWE-Prot\'eg\'e, a post-training framework that reframes software repair as an expert-prot\'eg\'e collaboration problem. In SWE-Prot\'eg\'e, an SLM remains the sole decision-maker while learning to selectively seek guidance from a strong expert model, recognize stalled states, and follow through on expert feedback. Our approach combines supervised fine-tuning on expert-augmented trajectories with agentic reinforcement learning that explicitly discourages degenerative looping and unproductive expert collaboration. We lightly post-train Qwen2.5-Coder-7B-Instruct to achieve 42.4% Pass@1 on SWE-bench Verified, a +25.4% improvement over the prior SLM state of the art, while using expert assistance sparsely (~4 calls per task and 11% of total tokens).

cs.SE↗

arXiv2019

Quantum gl(1|1) and tangle Floer homology

We identify the Grothendieck group of the tangle Floer dg algebra with a tensor product of certain $U_q(gl(1|1))$ representations. Under this identification, up to a scalar factor, the map on the Grothendieck group induced by the tangle Floer dg bimodule associated to a tangle agrees with the Reshetikhin-Turaev homomorphism for that tangle. We also introduce dg bimodules which act on the Grothendieck group as the generators $E$ and $F$ of $U_q(gl(1|1))$.

math.GT↗

arXiv2017

Super Kac-Moody 2-categories

We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.

math.RT↗

arXiv2017

Monoidal supercategories

In the literature, one finds several competing notions for the super (i.e., Z/2-graded) analog of a monoidal category. The goal of this paper is to clarify these definitions and the connections between them. We also discuss in detail the example of the odd Temperley-Lieb supercategory. In a forthcoming article, we will exploit the formalism developed here in order to define super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier.

math.RT↗

arXiv2016

The differential graded odd nilHecke algebra

We equip the odd nilHecke algebra and its associated thick calculus category with digrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum sl(2) at a fourth root of unity.

math.QA↗

arXiv2014

An odd categorification of quantum sl(2)

We define a 2-category that categorifies the covering Kac-Moody algebra for sl(2) introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a (Z x Z_2)-grading giving its Grothendieck group the structure of a free module over the group algebra of Z x Z_2. By specializing the Z_2-action to +1 or to -1, the construction specializes to an "odd" categorification of sl(2) and to a supercategorification of osp(1|2), respectively.

math.QA↗

arXiv2013

The Hopf algebra of odd symmetric functions

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.

math.QA↗

arXiv2011

The odd Littlewood-Richardson rule

In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via the odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model.

math.QA↗

arXiv2011

The odd nilHecke algebra and its diagrammatics

We introduce an odd version of the nilHecke algebra and develop an odd analogue of the thick diagrammatic calculus for nilHecke algebras. We graphically describe idempotents which give a Morita equivalence between odd nilHecke algebras and the rings of odd symmetric functions in finitely many variables. Cyclotomic quotients of odd nilHecke algebras are Morita equivalent to rings which are odd analogues of the cohomology rings of Grassmannians. Like their even counterparts, odd nilHecke algebras categorify the positive half of quantum sl(2).

math.QA↗