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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.

MSC:
 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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