#### Alladi Ramakrishnan Hall

#### Cellularity, Modular Representations, and Gram Matrices of a Class of Diagram Algebra

#### Karimilla Bi

##### IMSc

*1. Given a Gram matrix, is that possible to write the entries of the reduced Gram matrix obtained after applying the column and row operations from the corresponding diagrams for signed partition algebras introduced by S. Parvathi in 2011?.*

2. To compute the distinct eigenvalues of this reduced Gram matrix of the signed partition algebras?

In this connection, we realize that the signed partition algebras are tabular algebras and hence cellular algebra which was due to R. M. Green and P. P. Martin. As a consequence of cellularity, there exists a bilinear form and from which Gram matrix of signed partition

algebras can be defined.

The above work is also extended to the algebra of $\mathbb{Z}_2$-relations introduced by V. Kodiyalam, R. Srinivasan and V. S. Sunder in 2000 and partition algebras introduced by P. P. Martin and V. F. R. Jones in 1990.

Done