Terwilliger Algebra

The Terwilliger algebra of an association scheme at a point is the algebra generated by its Bose-Mesner algebra together with the diagonal dual idempotents , , ..., . The diagonal entry is 1 when and 0 otherwise, and every off-diagonal entry is 0. The Terwilliger algebra is also called the subconstituent algebra.

Block diagonalizations of Terwilliger algebras can reduce semidefinite programs arising in coding theory. For the binary Hamming scheme, Schrijver (2005) obtained upper bounds for binary codes and constant-weight binary codes that strengthen Delsarte's linear programming bound and can be computed in polynomial time as a function of the codeword length.

The Terwilliger algebra can depend on the base point , although base points in the same group orbit of the automorphism group give isomorphic algebras. Maleki and Razafimahatratra (2026) characterized when equals the centralizer algebra of the point stabilizer. They also characterized triply transitive quasi-thin association schemes and constructed new infinite families of triply transitive association schemes.

References

Maleki, R. and Razafimahatratra, A. S. "On the Terwilliger Algebras of Quasi-Thin Schurian Association Schemes." Elec. J. Combin. 33, No. 3, P3.34, 2026. https://doi.org/10.37236/14778.

Schrijver, A. "New Code Upper Bounds From the Terwilliger Algebra and Semidefinite Programming." IEEE Trans. Inform. Th. 51, 2859-2866, 2005. https://doi.org/10.1109/TIT.2005.851748.

Terwilliger, P. "The Subconstituent Algebra of an Association Scheme. I." J. Algebraic Combin. 1, 363-388, 1992. https://doi.org/10.1023/A:1022494701663.

Terwilliger Algebra

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