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PBW filtration and bases for irreducible modules in type \({\text{\textsf{A}}}_{n}\). (English) Zbl 1237.17011
The universal enveloping algebra \(U(\mathfrak{a})\) of a finite dimensional Lie algebra \(\mathfrak{a}\) has a natural filtration by monomial degrees. Let \(\mathfrak{g}=\mathfrak{sl}_n\) with a fixed standard triangular decomposition \(\mathfrak{g}=\mathfrak{n}_-\oplus\mathfrak{h}\oplus\mathfrak{n}_+\). For a dominant integral \(\lambda\in\mathfrak{h}^*\) let \(V(\lambda)\) be the corresponding simple finite dimensional \(\mathfrak{g}\)-module with a fixed primitive vector. Applying the components of the degree filtration of \(U(\mathfrak{n}_-)\) to the primitive vector induces a filtration on \(V(\lambda)\).
The main object of the study in the paper under review is the associated graded vector space \(\mathrm{gr}\,V(\lambda)\). The latter is a cyclic module over the symmetric algebra \(S(\mathfrak{n}_-)\) over \(\mathfrak{n}_-\) and hence has the form \(\mathfrak{n}_-/I(\lambda)\) for some ideal \(I(\lambda)\). The first main result of the paper gives a concrete description of \(I(\lambda)\). The second main result of the paper gives an explicit monomial basis of \(\mathrm{gr}\,V(\lambda)\), construction of which uses, in particular, combinatorics of Dyck paths.

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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