A Fock space approach to representation theory of osp(2|2n)

Cheng, Shun‐Jen; Wang, Weiqiang; Zhang, Ruibin

doi:10.1007/s00031-006-0040-5

Cited by 9 publications

(9 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Partly due to the time constraint of the lectures, we have left out many interesting topics on super representation theory. We refer to [BL, J] (and more recently [SZ]) for finite-dimensional irreducible characters of atypicality one, to [BKN, DS, Ma, Mu, Pe, PS] for geometric approaches, to [Br2,CWZ2] for further development of the Fock space approach of Brundan for the queer Lie superalgebra q(n) and for osp(2|2n), to [CK,CKW,CZ1,Ger,San,Sva,Zou] for some cohomological aspects, to [BrK, SW, WZ] for prime characteristic, to [JHKT, Su] for related combinatorial structures; also see [Gor, KW, Naz] for additional work on Lie superalgebras.…”

mentioning

confidence: 99%

Dualities and Representations of Lie Superalgebras

Cheng

Wang

2012

Graduate Studies in Mathematics

Self Cite

283

405

View full text Add to dashboard Cite

We explain how Lie superalgebras of types gl and osp provide a natural framework generalizing the classical Schur and Howe dualities. This exposition includes a discussion of super duality, which connects the parabolic categories O between classical Lie superalgebras and Lie algebras. Super duality provides a conceptual solution to the irreducible character problem for these Lie superalgebras in terms of the classical Kazhdan-Lusztig polynomials.

show abstract

mentioning

confidence: 99%

Dualities and Representations of Lie Superalgebras

Cheng

Wang

2012

Graduate Studies in Mathematics

Self Cite

283

405

View full text Add to dashboard Cite

show abstract

“…We now show that for a Type-I Lie superalgebra the category F is a highest weight category in the sense of [8]. This was proven for gl(m|n) by Brundan [4] (see also [13,Section 3.6]), and is implicit for osp(2|2n) in the work of Cheng, Wang, and Zhang [7]. We provide a general proof which includes both of these as cases.…”

Section: Filtrations By Kac Supermodulesmentioning

confidence: 76%

“…However, in [17,Theorem 1] it is asserted that for a basic classical Lie superalgebra such as osp(3|2), K(λ) = L(λ) if and only if λ is typical. This counterexample may be known to experts but we could not find a suitable reference in the literature (we have since been informed that the authors of [7] were aware of this example). We expect that the result in [17] is correct for Type-I Lie superalgebras.…”

Section: Type-ii Lie Superalgebrasmentioning

confidence: 99%

Cohomology and support varieties for Lie superalgebras II

Boe

Kujawa

Nakano

2008

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

In [2] (Preprint, 2006, arXiv:math.RT/0609363) the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra, and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties of Kac supermodules for Type-I Lie superalgebras and of the simple supermodules for gl(m|n). The latter result verifies our earlier conjecture for gl(m|n). In our investigation we also delineate several of the major differences between Type-I versus Type-II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

show abstract

“…There exist a number of other interesting recent works which emphasize the Lie superalgebraic and combinatorial side of the picture (see, e.g., [4][5][6]). …”

Section: Page 4 Of 73 Huckleberry Et Al Complex Analysis and Its Synmentioning

confidence: 99%

“…Our explicit formula for I(t) looks exactly like a classical Weyl formula and is derived in terms of the roots of the Lie superalgebra g and the Weyl group W. Let us state this formula for K = O N , USp N without going into the details of the -positive even and odd roots + ,0 and + ,1 and the Weyl group W (see Sect. 5.3 for precise formulas). If W is the isotropy subgroup of W fixing the highest weight = N , then…”

Section: Introductionmentioning

confidence: 99%

Haar expectations of ratios of random characteristic polynomials

Huckleberry

Püttmann

Zirnbauer

2016

Complex Analysis and its Synergies

View full text Add to dashboard Cite

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O N , SO N , and USp N . To that end, we start from the Clifford-Weyl algebra in its canonical realization on the complex A V of holomorphic differential forms for a C-vector space V 0 . From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V 0 . Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp ⊂ osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V 0 = C n ⊗ C N where C N is equipped with the standard K-representation, and focus on the subspace A K V of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K ) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl-group invariance, certain weight constraints, and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way, we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N ≥ 1. This completes earlier work of Conrey, Farmer, and Zirnbauer for the case of U N .

show abstract

A Fock space approach to representation theory of osp(2|2n)

Cited by 9 publications

References 14 publications

Dualities and Representations of Lie Superalgebras

Dualities and Representations of Lie Superalgebras

Cohomology and support varieties for Lie superalgebras II

Haar expectations of ratios of random characteristic polynomials

Contact Info

Product

Resources

About