#### Presentation Title

Representations of the Temperley-Lieb Algebra

#### Faculty Mentor

Wade Bloomquist

#### Start Date

18-11-2017 10:00 AM

#### End Date

18-11-2017 11:00 AM

#### Location

BSC-Ursa Minor 107

#### Session

Poster 1

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

The Temperley-Lieb algebra, *TL _{n}(δ)*, is a vector space with a basis of diagrams that are boxes with

*n*marked points on the bottom and

*n*on the top, connected by non-crossing lines. Diagrams are multiplied by stacking one atop another, potentially creating closed loops that may be removed through multiplication by the scalar

*δ*. One representation of this algebra is the vector space having a basis of half diagrams, also called link diagrams. Half diagrams have

*n*marked points on an interval, some connected by curves without intersection, and others with defects, or through lines. These diagrams, when acted on by elements of

*T L*, construct matrices. A different representation of

_{n}(δ)*TL*arises from statistical mechanics and the study of magnetic systems in lattices, called the spin chain representation. We have studied how to bridge the gap between these differing approaches to the representation theory of the Temperley-Lieb algebra. Through our construction of explicit intertwining operators — isomorphisms that relate the two vector spaces and commute with the action of

_{n}(δ)*TL*— we can examine the spin chain representation entirely through diagrammatic means.

_{n}(δ)#### Summary of research results to be presented

In his 2015 paper, Jim de Groot conjectured that "On the level of vector spaces, *(C ^{2})*

^{⊗n}is isomorphic to ⊕

*." Using some lemmas based in combinatorics, we have shown this to be true. In the process of finding an explicit isomorphism that is also an intertwiner, we also proved that de Groot's intertwiner*

_{i=0}^{⌊n/2⌋}(n + 1 − 2i)S(n, i)*Ω _{n,p} : S(n, p) → (C^{2})*

^{⊗n}given by

*Ω*

_{n,p}(w) = π_{(j,j′)∈ψ(w)}T(j,j′)(v_{+}^{⊗n}

*)*is injective. This allows us to construct a larger intertwiner that can isomorphically map the spin chains to half diagrams.

Representations of the Temperley-Lieb Algebra

BSC-Ursa Minor 107

The Temperley-Lieb algebra, *TL _{n}(δ)*, is a vector space with a basis of diagrams that are boxes with

*n*marked points on the bottom and

*n*on the top, connected by non-crossing lines. Diagrams are multiplied by stacking one atop another, potentially creating closed loops that may be removed through multiplication by the scalar

*δ*. One representation of this algebra is the vector space having a basis of half diagrams, also called link diagrams. Half diagrams have

*n*marked points on an interval, some connected by curves without intersection, and others with defects, or through lines. These diagrams, when acted on by elements of

*T L*, construct matrices. A different representation of

_{n}(δ)*TL*arises from statistical mechanics and the study of magnetic systems in lattices, called the spin chain representation. We have studied how to bridge the gap between these differing approaches to the representation theory of the Temperley-Lieb algebra. Through our construction of explicit intertwining operators — isomorphisms that relate the two vector spaces and commute with the action of

_{n}(δ)*TL*— we can examine the spin chain representation entirely through diagrammatic means.

_{n}(δ)