#### Alladi Ramakrishnan Hall

#### Thesis Defense: On bases for local Weyl modules in type A

#### B Ravinder

##### TIFR Mumbai

*We study the Chari-Pressley-Loktev bases for local Weyl modules of the current algebra sl_{r+1}[t]:*

1. As convenient parametrizing sets of these bases, we introduce the notion of partition overlaid patterns (POPs).

2. 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.

3. We prove the "stability" of Chari-Pressley-Loktev bases with respect to inclusions of local Weyl modules in the case r=1 and state it as a conjecture for r>1.

4. In order to state the stability conjecture, we establish a certain bijection between colored partitions and POPs, which is of interest in itself.

5. The grade zero basis elements form a basis for the corresponding irreducible representation space of sl_{r+1}. We prove a triangular relationship

between this basis and the classical Gelfand-Tsetlin basis.

Done