Brauer Algebra Research Articles

Critical spin chains and loop models with $PSU(n)$ symmetry

Starting with the Ising model, statistical models with global symmetries provide fruitful approaches to interesting physical systems, for example percolation or polymers. These include the O(n)O(n) model (symmetry group O(n)O(n)) and the Potts model (symmetry group S_QSQ). Both models make sense for n,Q∈ \mathbb{C}n,Q∈ℂ and not just n,Q∈ \mathbb{N}n,Q∈ℕ, and both give rise to a conformal field theory in the critical limit. Here, we study similar models based on the group PSU(n)PSU(n). We focus on the two-dimensional case, where the models can be described either as gases of non-intersecting orientable loops, or as alternating spin chains. This allows us to determine their spectra either by computing a twisted torus partition function, or by studying representations of the walled Brauer algebra. In the critical limit, our models give rise to a CFT that exists for any n∈\mathbb{C}n∈ℂ and has a global PSU(n)PSU(n) symmetry. Its spectrum is similar to those of the O(n)O(n) and Potts CFTs, but a bit simpler. We conjecture that the O(n)O(n) CFT is a \mathbb{Z}_2ℤ2 orbifold of the PSU(n)PSU(n) CFT, where \mathbb{Z}_2ℤ2 acts as complex conjugation.

Linear Programming with Unitary-Equivariant Constraints

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a dp+q-dimensional matrix variable that commutes with U⊗p⊗U¯⊗q, for all U∈U(d). Solving such problems naively can be prohibitively expensive even if p+q is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

Idempotents and homology of diagram algebras

This paper provides a systematization of some recent results in homology of algebras. Our main theorem gives criteria under which the homology of a diagram algebra is isomorphic to the homology of the subalgebra on diagrams having the maximum number of left-to-right connections. From this theorem, we deduce the ‘invertible-parameter’ cases of the Temperley–Lieb and Brauer results of Boyd–Hepworth and Boyd–Hepworth–Patzt. We are also able to give a new proof of Sroka’s theorem that the homology of an odd-strand Temperley–Lieb algebra vanishes, as well as an analogous result for Brauer algebras and an interpretation of both results in the even-strand case. Our proofs are relatively elementary: in particular, no auxiliary chain complexes or spectral sequences are required. We briefly discuss the relationship to cellular algebras in the sense of Graham–Lehrer.

Positive maps from the walled Brauer algebra

We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.

Morita equivalences on Brauer algebras and BMW algebras of simply-laced types

The Morita equivalences of classical Brauer algebras and classical Birman–Murakami–Wenzl (BMW) algebras have been well studied. Here, we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and BMW algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of Coxeter groups, respectively.

Representation stability for diagram algebras

Representation stability for diagram algebras

On the Grothendieck Ring of the Sequence of Walled Brauer Algebras

In this paper, we provide a diagrammatic approach to study the branching rules for cell modules on a sequence of walled Brauer algebras. This approach also allows us to calculate the structure constants of multiplication over the Grothendieck ring of the sequence.

Spaces of states of the two-dimensional $O(n)$ and Potts models

We determine the spaces of states of the two-dimensional O(n)O(n) and QQ-state Potts models with generic parameters n,Q\in \mathbb{C}n,Q∈ℂ as representations of their known symmetry algebras. While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the O(n)O(n) model, and the symmetric group S_QSQ for the QQ-state Potts model. We do this by two independent methods. First we compute the twisted torus partition functions of the models at criticality. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the O(n)O(n) model, we decompose representations of the Brauer algebra into representations of its unoriented Jones-Temperley-Lieb subalgebra. For the QQ-state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on four-point functions of the corresponding CFTs.

Discriminants of quantized walled Brauer algebras

Discriminants of quantized walled Brauer algebras

Walled Klein-4 Brauer algebras

The new class of diagram algebras known as walled Klein-$4$ Brauer algebras, denoted by $\overrightarrow{\overrightarrow{D}}_{r,s}(l)$, where $r,s \in \mathds{N}$ and $l \in K$, is an indeterminate, is studied in this paper. The walled klein-$4$ Brauer algebras are explained in terms of generators and relations. The indexing set of the simple modules of the walled Klein-$4$ Brauer algebras was described. We established that walled Klein-$4$ Brauer algebras are iterated inflations of the group algebra of the group $(\mathds{Z}_{2}\times \mathds{Z}_{2})\wr \mathbf{S}_{r} \times (\mathds{Z}_{2}\times \mathds{Z}_{2})\wr \mathbf{S}_{s}$, and we concluded that $\overrightarrow{\overrightarrow{D}}_{r,s}(l)$ is cellular as a consequence.

Efficient Multi Port-Based Teleportation Schemes

In this manuscript we analyse generalised port-based teleportation (PBT) schemes, allowing for transmitting more than one unknown quantum state (or a composite quantum state) in one go, where the state ends up in several ports at Bob’s side. We investigate the efficiency of our scheme discussing both deterministic and probabilistic case, where parties share maximally entangled states. It turns out that the new scheme gives better performance than various variants of the optimal PBT protocol used for the same task. All the results are presented in group-theoretic manner depending on such quantities like dimensions and multiplicities of irreducible representations in the Schur-Weyl duality. The presented analysis was possible by considering the algebra of permutation operators acting on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n-$ </tex-math></inline-formula> fold tensor product of the Hilbert space with finite dimension, we present construction of the respective irreducible matrix representations, which are in fact matrix irreducible representations of the Walled Brauer Algebra. I turns out that the introduced formalism, and symmetries beneath it, appears in many aspects of theoretical physics and mathematics - theory of anti ferromagnetism, aspects of gravity theory or in the problem of designing quantum circuits for special task like for example inverting an unknown unitary.

Compact Schur-Weyl duality: Real Lie groups and the cyclotomic Brauer algebra

We show that the centraliser of the maximal compact subgroup of the real groups O ( p , q ) and S p 2 n ( R ) acting on tensors of their standard representation is isomorphic to the level 2 cyclotomic Brauer algebra with certain parameters in C . We also prove that for S p 2 n ( R ) this cyclotomic Brauer algebra is Morita equivalent to a direct sum of walled Brauer algebras.

Handlebody diagram algebras

In this paper we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

A Combinatorial Translation Principle and Diagram Combinatorics for the Symplectic Group

Let k be an algebraically closed field of characteristic p > 2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups.

Manin matrices of type C: Multi-parametric deformation

We constructed a multi-parametric deformation of the Brauer algebra representation related with the symplectic Lie algebras. The notion of Manin matrix of type C was generalised to the case of the multi-parametric deformation by using this representation and corresponding quadratic algebras. We derived pairing operators for these quadratic algebras and minors for the considered Manin matrices. The rank of pairing operators and dimensions of components of quadratic algebras were calculated.

A COMBINATORIAL TRANSLATION PRINCIPLE AND DIAGRAM COMBINATORICS FOR THE GENERAL LINEAR GROUP

Let k be an algebraically closed field of characteristic p > 0. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the general linear group over k in terms of cap diagrams under the assumption that p is bigger than the greatest hook length in the partitions involved. Then we introduce and study the rational Schur functor from a category of GLn-modules to the category of modules for the walled Brauer algebra. As a corollary, we obtain the decomposition numbers for the walled Brauer algebra when p is bigger than the greatest hook length in the partitions involved. This is a sequel to an earlier paper on the symplectic group and the Brauer algebra.

On a Class of Orthogonal-Invariant Quantum Spin Systems on the Complete Graph

AbstractWe study a two-parameter family of quantum spin systems on the complete graph, which is the most general model invariant under the complex orthogonal group. In spin $S=\frac {1}{2}$ it is equivalent to the XXZ model, and in spin $S=1$ to the bilinear-biquadratic Heisenberg model. The paper is motivated by the work of Björnberg, whose model is invariant under the (larger) complex general linear group. In spin $S=\frac {1}{2}$ and $S=1$ we give an explicit formula for the free energy for all values of the two parameters, and for spin $S&amp;gt;1$ for when one of the parameters is non-negative. This allows us to draw phase diagrams and determine critical temperatures. For spin $S=\frac {1}{2}$ and $S=1$, we give the left and right derivatives as the strength parameter of a certain magnetisation term tends to zero, and we give a formula for a certain total spin observable, and heuristics for the set of extremal Gibbs states in several regions of the phase diagrams, in the style of a recent paper of Björnberg, Fröhlich, and Ueltschi. The key technical tool is expressing the partition function in terms of the irreducible characters of the symmetric group and the Brauer algebra. The parameters considered include, and go beyond, those for which the systems have probabilistic representations as interchange processes.

The Manhattan and Lorentz Mirror Models: A Result on the Cylinder with Low Density of Mirrors

We study the Manhattan and Lorentz mirror models on an infinite cylinder of finite even width n, with the mirror probability p satisfying p<Cn^{-1}, C a constant. We show that the maximum height along the cylinder reached by a walker is order p^{-2}. We observe an algebraic structure, which helps organise our argument. The models on the cylinder can be thought of as Markov chains on the Brauer (in the Mirror case) or Walled Brauer (in the Manhattan case) algebra, with the transfer matrix given by multiplication by an element of the algebra.

The homology of the Brauer algebras

This paper investigates the homology of the Brauer algebras, interpreted as appropriate {{,mathrm{Tor},}}-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter delta of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when delta is not invertible, this isomorphism still holds in a range of degrees that increases with n.

The finite dimensional irreducible modules for affine walled Brauer algebras

• d -semi-admissible or admissible conditions. • The cyclotomic walled Brauer algebra with non-admissible parameters can be deduced from that with admissible parameters. • Classify the finite dimensional irreducible modules for affine walled Brauer algebras. We classify the finite dimensional irreducible modules for affine walled Brauer algebras over an algebraically closed field with arbitrary characteristic.