Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A A is a quasi-hereditary algebra with a simple preserving duality and T T is a faithful tilting A A -module, then A A has the double centralizer property with respect to T T . This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T T over A A for which A = E n d E n d A ( T ) ( T ) A=End_{End_A(T)}(T) . As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra S K s y ( m , n ) S_K^{sy}(m,n) and the Brauer algebra B n ( − 2 m ) \mathfrak {B}_n(-2m) on the space of dual partially harmonic tensors under certain condition.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

Schur–Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers $$U^{\otimes t}$$ U ⊗ t of all unitaries $$U\in U(d)$$ U ∈ U ( d ) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Dominant and global dimension of blocks of quantised Schur algebras

Group algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

Jucys–Murphy elements of partition algebras for the rook monoid

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra [Formula: see text] and the subalgebra [Formula: see text] of the partition algebra [Formula: see text] acting on [Formula: see text]. In this paper, we consider a subalgebra [Formula: see text] of [Formula: see text] such that there is Schur–Weyl duality between the actions of [Formula: see text] and [Formula: see text] on [Formula: see text]. This paper studies the representation theory of partition algebras [Formula: see text] and [Formula: see text] for rook monoids inductively by considering the multiplicity free tower [Formula: see text] Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of [Formula: see text] and [Formula: see text]. Also, we describe the Jucys–Murphy elements of [Formula: see text] which play a central role in the demonstration of the actions of Jucys–Murphy elements of [Formula: see text] and [Formula: see text].

Schur–Weyl duality and the product of randomly-rotated symmetries by a unitary Brownian motion

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.

On the representation theory of the infinite Temperley–Lieb algebra

We begin the study of the representation theory of the infinite Temperley–Lieb algebra. We fully classify its finite-dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TL[Formula: see text] that generalizes to give extensions of TL[Formula: see text] representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.