In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_{\mathit{\lambda }}(\mathit{x};q)$ in some special cases. We associate a weighted graph $\mathrm{\Pi }$ to $\mathit{\lambda }$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_{\mathit{\lambda }}(\mathit{x};q)$ whenever $\mathrm{\Pi }$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.

Keywords: Charge, chromatic symmetric function, cocharge, Hall--Littlewood polynomial, jeu de taquin, LLT polynomial, interval graph, Schur function, Schur-positive, symmetric function.

Mathematics Subject Classifications: 05E05, 05E10, 05C15

We prove that two horizontal-strip LLT polynomials are equal if the associated weighted graphs defined by the author in a previous paper are isomorphic. This provides a sufficient condition for equality of horizontal-strip LLT polynomials and yields a well-defined LLT polynomial indexed by a weighted graph. We use this to prove some new relations between LLT polynomials and we explore a connection with extended chromatic symmetric functions.

Mathematics Subject Classifications: 05E05, 05E10, 05C15

Keywords: Chromatic symmetric function, LLT polynomial, Hall-Littlewood polynomial, interval graph, Schur function, weighted graph

In studying Hoodia patent law struggles, I am interested in how sovereign power, in the service of neoliberal bioeconomies, values some forms of knowledge over others. I examine how techniques of governmentality such as patent law, benefit sharing contracts, bioprospecting permits, and prior informed consent agreements are being used to structure inequitable forms of citizenship based upon whose knowledge and intellectual labor matters more to the neoliberal project of the nation-state. In particular, I ask how relevant social actors make claims for rights, benefits, and protection under the law based upon a vulnerability to their processes and ways of knowing in order to participate more fully within global market economies. In addition, I examine how social actors articulate, position, and rework concepts of nature and culture as they describe their practices related to the plant in order to secure rights under patent law and benefit sharing legislation.