Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation

Modified Villain formulation of Abelian Chern-Simons theory

We formulate U(1)k Chern-Simons theory on a Euclidean spacetime lattice using the modified Villain approach. Various familiar aspects of continuum Chern-Simons theory such as level quantization, framing, the discrete 1-form symmetry and its ’t Hooft anomaly, as well as the electric charge of monopole operators are manifest in our construction. The key technical ingredient is the cup product and its higher generalizations on the (hyper-)cubic lattice, which recently appeared in the literature. All unframed Wilson loops are projected out by a peculiar subsystem symmetry, leaving topological, ribbonlike Wilson loops which have the correct correlation functions and topological spins expected from the continuum theory. Our action can be obtained from a new definition of the theta term in four dimensions which improves upon previous constructions within the modified Villain approach. This bulk action coupled to background fields for the 1-form symmetry is given by the Pontryagin square, which provides anomaly inflow directly on the lattice

Canonical quantization of lattice Chern-Simons theory

We discuss the canonical quantization of $U(1)_k$ Chern-Simons theory on a spatial lattice. In addition to the usual local Gauss law constraints, the physical Hilbert space is defined by 1-form gauge constraints implementing the compactness of the $U(1)$ gauge group, and (depending on the details of the spatial lattice) non-local constraints which project out unframed Wilson loops. Though the ingredients of the lattice model are bosonic, the physical Hilbert space is finite-dimensional, with exactly $k$ ground states on a spatial torus. We quantize both the bosonic (even level) and fermionic (odd level) theories, describing in detail how the latter depends on a choice of spin structure.Comment: 25 pages, 6 figure