[Paper Review] Modified Traces on Deligne's Category Rep(S_{t})

Jonathan Comes, Jonathan R. Kujawa|arXiv (Cornell University)|Mar 10, 2011

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper establishes the existence of a unique nontrivial family of modified trace functions in Deligne's category Rep(Sₜ), proving that when t is a nonnegative integer, the category admits modified traces despite not being abelian. The authors introduce a graded variant of the category, lift the modified traces to this setting, and show they recover the writhe invariant of framed knots via a q-deformation of the braiding and twist structures.

ABSTRACT

Deligne has defined a category which interpolates among the representations of the various symmetric groups. In this paper we show Deligne's category admits a unique nontrivial family of modified trace functions. Such modified trace functions have already proven to be interesting in both low-dimensional topology and representation theory. We also introduce a graded variant of Deligne's category, lift the modified trace functions to the graded setting, and use them to recover the well-known invariant of framed knots known as the writhe.

Motivation & Objective

• To investigate the existence and structure of modified trace functions in Deligne's category Rep(Sₜ), which interpolates representations of symmetric groups.
• To address the gap in understanding when modified traces exist, especially in nonabelian categories, by studying Rep(Sₜ) as a nonabelian example.
• To construct a graded version of Deligne's category to lift modified traces and connect them to topological invariants like the writhe.
• To demonstrate that modified traces in the graded setting recover the classical writhe invariant of framed knots via a q-deformed braiding.

Proposed method

• Define a graded variant of Deligne's category, denoted gRep₀(Sₜ)ₚ, by introducing a ℤ-grading on the objects and morphisms.
• Construct a tensor functor F: gRep(Sₜ)ₚ → Rep(Sₜ) that forgets the grading and maps objects [a,b] to [a+b] and morphisms identically.
• Lift modified traces from Rep(Sₜ) to the graded category using the functor F, showing that if F(V) admits an ambidextrous trace, then V does too.
• Use the graded braiding and twist structures, where c_{V,W} is scaled by q^{rs} and θ_V by q^{r²}, to relate the categorical structure to topological invariants.
• Compute the invariant of an oriented framed knot K labeled by M_{a,b} as q^{(a−b)²ω}, where ω is the writhe of K.
• Verify that the lifted trace structure commutes with evaluation, coevaluation, and braiding, preserving the ambidextrous trace condition.

Experimental results

Research questions

• RQ1 Does Deligne's category Rep(Sₜ) admit nontrivial modified trace functions, and if so, under what conditions?
• RQ2 Can modified traces be lifted from the ungraded to a graded version of Deligne's category?
• RQ3 What topological invariant emerges from the modified trace in the graded setting, particularly for framed knots?
• RQ4 How does the q-deformation of the braiding and twist in the graded category relate to known invariants like the writhe?
• RQ5 Is the existence of modified traces possible in nonabelian ribbon categories, as in Rep(Sₜ) when t is a nonnegative integer?

Key findings

• When t is a nonnegative integer, Deligne's category Rep(Sₜ) admits a unique nontrivial family of modified trace functions.
• The category Rep(Sₜ) is the first known example of a nonabelian ribbon category that supports modified traces, despite not being abelian.
• The graded category gRep(Sₜ)ₚ admits a tensor functor F to Rep(Sₜ) that preserves duality, associativity, and unit constraints.
• Modified traces in Rep(Sₜ) lift to the graded setting via the functor F, ensuring that objects like M_{a,b} in gRep(Sₜ)ₚ admit ambidextrous traces.
• For a framed knot K labeled by M_{a,b}, the invariant is q^{(a−b)²ω}, where ω is the writhe of K, recovering the classical writhe invariant.
• The construction shows that the writhe invariant arises naturally from the q-deformed braiding and twist in the graded category, valid when q is not a root of unity.