Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra q(n)

By Jonathan Brundan. 2002

251 citations recorded by the source corpus.

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak q(n)$ over $\C$ was solved in 1996 by I. Penkov and V. Serganova. In this article, we give a different approach relating the character problem to canonical bases of the quantized enveloping algebra $U_q(\mathfrak b_{\infty})$. We also formulate for the first time a conjecture for the characters of the infinite dimensional irreducible representations in the analogue of category $\mathcal O$ for the Lie superalgebra $\mathfrak{q}(n)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0f3c8f9f218397b5085c5c370f66ed2668bbf7ac

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Citations in the corpus

Modular representations of the supergroup Q(n), I (2003, methodology): “These flag superschemes, and their analogues G/P for arbitrary parabolic subgroups of G, were introduced originally by Manin [12, 13], and have already played a fundamental role in the work of Penkov and Serganova [18, 19, 20] on the representation theory of Q(n) over C; see also [1].”
Dual canonical bases, quantum shuffles and q-characters (2002, background): “Since then these connections have been studied by several authors and other similar correspondences have been discovered [A, LT2, VV, Gr, BK, LNT, B1 , B2].”
The brundan-kazhdan-lusztig conjecture for general linear lie superalgebras (2012, background): “nal paper [Br1] Brundan formulated an elegant conjecture on the characters for the irreducible modules and tilting modules in the full BGG category Om|n of gl(m|n)-modules for the ﬁrst time (also see [Br3] for a similar formulation for the Lie superalgebra q(n)). Brundan’s formulation was in terms o”
Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras (2009, methodology): “imilar type formulae were conjectured in [18] for aﬃne superalgebras. In 1996, the second author [24] solved the problem completely for gl(m,n) using the geometry of ﬂag supervarieties. Later Brundan [3] solved this problem by a diﬀerent, purely algebraic, method and discovered a remarkable connecti”
Highest weight categories arising from Khovanov's diagram algebra IV: the general linear supergroup (2009, background): “As explained in [ B1 , §4-e], the category of finite dimensional G-modules decomposes as F⊕�F, where F = F(m|n) is the full subcategory consisting of the modules all of whose composition factors are isomorphic to L(λ)’s for various λ ∈ X+(T), and �F is the image of Funder �. Note also that F is clos”
Kac–Moody Superalgebras and Integrability (2011, unknown): “Since in our case the defect is 1, our formulae are easier than those for general finite-dimensional superalgebras (see [4, 15]).”
Highest weight categories arising from Khovanov's diagram algebra II: Koszulity (2008, methodology): “The following theorem is the central result of the section; the proof here is based on [ Br , Lemma 4.49].”
Generalized trace and modified dimension functions on ribbon categories (2010, methodology): “ Lλ ⊗ V is a polynomial representation, it is completely reducible. Taken together this implies Lµ is a direct summand of Lλ ⊗ V and, hence, ILµ ⊆ IL λ . Furthermore, in the crystal graph language of [15, 41] there is a directed edge from Lλ to Lµ colored with some integer a∈ Zand Lµ = FaLλ. Since F”
A new approach to Kazhdan-Lusztig theory of type ${\bf B}$ via quantum symmetric pairs (2013, methodology): “ bar involution ψı : V⊗m → V⊗m is given by ψı = Υψ, where ψ: V⊗m → V⊗m is a bar involution of type A. The following compatibility of the bar involutions in the type Asetting is well known (see, e.g., [Br1]) (Here we note that our ı-anti-dominant condition is stronger than the type Aanti-dominant con”
Character and dimension formulae for general linear superalgebra (2004, methodology): “Serganova’s work was further developed in the papers [18, 17, 2 , 3, 4]. Particularly important is the work of Brundan, developed a very practicable algorithm for computing the generalized Kazhdan-Lusztig polynomials, by using quantum”
Grothendieck rings of basic classical Lie superalgebras (2007, methodology): “[18] V. Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra gl(m|n), Selecta Math.”