This paper is a study of the bases introduced by Chari-Loktev in [1] for local Weyl modules of the current algebra associated to a special linear Lie algebra. Partition overlaid patterns, POPs for shortwhose introduction is one of the aims of this paper-form convenient parametrizing sets of these bases. They play a role analogous to that played by (Gelfand-Tsetlin) patterns in the representation theory of the special linear Lie algebra.The notion of a POP leads naturally to the notion of area of a pattern. We observe that there is a unique pattern of maximal area among all those with a given bounding sequence and given weight. We give a combinatorial proof of this and discuss its representation theoretic relevance.We then state a conjecture about the "stability", i.e., compatibility in the long range, of Chari-Loktev bases with respect to inclusions of local Weyl modules. In order to state the conjecture, we establish a certain bijection between colored partitions and POPs, which may be of interest in itself. The stability conjecture has been proved in [6] in the rank one case.2010 Mathematics Subject Classification. 17B67 (05E10). Key words and phrases. Gelfand-Tsetlin pattern, Chari-Loktev basis, Demazure module, partition overlaid pattern, POP, area of a pattern, stability of Chari-Loktev bases.The second named author acknowledges support from CSIR under the SPM doctoral fellowship scheme. The first and third named authors acknowledge support from DAE under a XII plan project. 1 One of our aims in the present paper is to clarify the parametrizing set of this Chari-Loktev basis. Towards this end we introduce the notion of a partition overlaid pattern or POP (see §2.8). Dominant integral weights for g = sl r+1 may be identified with non-increasing sequences of non-negative integers of length r + 1 with the last element of the sequence being 0. Gelfand-Tsetlin patterns (or just patterns-see §2 for the precise definition) with bounding sequence (corresponding to) λ parametrize the Gelfand-Tsetlin basis for V (λ). Analogously, as we show in §4.6, POPs with bounding sequence λ parametrize the Chari-Loktev basis for W (λ). The weight of the underlying pattern of a POP equals the h-weight of the corresponding basis element and the number of boxes in the partition overlay determines the grade.The notion of a POP leads naturally to the notion of the area of a pattern (see §2.3). For a weight µ of V (λ), it turns out that the piece of highest grade in the µ-weight space of the local Weyl module W (λ) is one dimensional (the h-weights of W (λ) are precisely those of V (λ)). We give a representation theoretic proof of this fact in §4. This suggests-even proves, albeit circuitously-that there must be a unique pattern of highest area among all those with bounding sequence λ and weight µ. We give a direct, elementary, and purely combinatorial proof of this in §3.In §6, we state a conjecture about the "stability" of the Chari-Loktev basis. To describe what is meant by stability, let θ be the highest root of g. We then have nat...

Abstract. We prove stability of the Chari-Pressley-Loktev bases for natural inclusions of local Weyl modules of the current algebra sl2 [t]. These modules being known to be Demazure submodules in the level 1 representations of the affine Lie algebra sl2, we obtain, by passage to the direct limit, bases for the level 1 representations themselves.

Abstract. Let g be a finite-dimensional complex simple Lie algebra with highest root θ. Given two non-negative integers m, n, we prove that the fusion product of m copies of the level one Demazure module D(1, θ) with n copies of the adjoint representation ev 0 V (θ) is independent of the parameters and we give explicit defining relations. As a consequence, for g simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of θ.

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$ and let $\mathfrak{g}[t]$ be the corresponding current algebra. In this paper, we consider the $\mathfrak{g}[t]$-stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra $\widehat{\mathfrak{g}}$. We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of $\theta$ is a generalized Demazure module, and also give defining relations. This also shows that the fusion product of such Demazure modules is independent of the chosen parameters. As a consequence we obtain generators and relations for certain types of generalized Demazure modules. We also establish a connection with the modules defined by Chari and Venkatesh.Comment: 24 pages; minor revision

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