Introduction to Representation Theory

Etingof, Pavel; Golberg, Oleg; Hensel, Sebastian; Liu, Tiankai; Schwendner, Alex; Vaintrob, Dmitry; Yudovina, Elena

doi:10.1090/stml/059

Cited by 63 publications

(75 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence, if [C]J is multiplicitly-free, thenm(C, λ) =m(C, λ) 2 for all λ ∈ Λ and so C (2) = C∩C −1 . On the other hand, if C (2) = C∩C −1 , then λ∈Λm (C, λ) = λ∈Λm (C, λ) 2 and som(C, λ) ∈ {0, 1} for all λ ∈ Λ. Thus, up to signs, the numbers c w,λ are the leading coefficients of character values as defined by Lusztig [14].…”

Section: The Ringjmentioning

confidence: 99%

See 1 more Smart Citation

Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Geck

2014

Journal of Algebra

View full text Add to dashboard Cite

Abstract. Let W be a finite Coxeter group. It is well-known that the number of involutions in W is equal to the sum of the degrees of the irreducible characters of W . Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of W into Kazhdan-Lusztig cells. The proof uses a generalisation of the Frobenius-Schur indicator to symmetric algebras, which may be of independent interest.

show abstract

Section: The Ringjmentioning

confidence: 99%

“…. , C r be the leftJ-cells which are contained in C. Then, clearly, C (2) is the union of the sets of involutions in C 1 , . .…”

Section: The Ringjmentioning

confidence: 99%

Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Geck

2014

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…It can be shown that (see Theorem 4.75, [5], for example) this representation is independent from the choice of x ∈ [x]. Also, if {[x 1 ], .…”

Section: Semi-direct Product Of Finite Abelian Groups and Charactersmentioning

confidence: 99%

Symmetry classes of tensors and semi-direct product of finite abelian groups

Rodtes

Chimla

2017

Linear and Multilinear Algebra

View full text Add to dashboard Cite

Abstract. In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite groups yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.

show abstract

“…This section will briefly provide some definitions and the results used in this work. For more details the reader can refer to textbooks like [37,52]. Let G denote a finite group, V some finite-dimensional complex vector space.…”

Section: Representation Theorymentioning

confidence: 99%

Efficient unitarity randomized benchmarking of few-qubit Clifford gates

2019

View full text Add to dashboard Cite

Unitarity randomized benchmarking (URB) is an experimental procedure for estimating the coherence of implemented quantum gates independently of state preparation and measurement errors. These estimates of the coherence are measured by the unitarity. A central problem in this experiment is relating the number of data points to rigorous confidence intervals. In this work we provide a bound on the required number of data points for Clifford URB as a function of confidence and experimental parameters. This bound has favorable scaling in the regime of near-unitary noise and is asymptotically independent of the length of the gate sequences used. We also show that, in contrast to standard randomized benchmarking, a nontrivial number of data points is always required to overcome the randomness introduced by state preparation and measurement errors even in the limit of perfect gates. Our bound is sufficiently sharp to benchmark small-dimensional systems in realistic parameter regimes using a modest number of data points. For example, we show that the unitarity of single-qubit Clifford gates can be rigorously estimated using few hundred data points under the assumption of gate-independent noise. This is a reduction of orders of magnitude compared to previously known bounds.

show abstract

Introduction to Representation Theory

Cited by 63 publications

References 0 publications

Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Kazhdan–Lusztig cells and the Frobenius–Schur indicator

Symmetry classes of tensors and semi-direct product of finite abelian groups

Efficient unitarity randomized benchmarking of few-qubit Clifford gates

Contact Info

Product

Resources

About