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Terwilliger Algebra

The Terwilliger algebra T(x) of an association scheme at a point x is the algebra generated by its Bose-Mesner algebra together with the diagonal dual idempotents E_0^*(x), E_1^*(x), ..., E_d^*(x). The diagonal entry (E_i^*)_(yy) is 1 when (x,y) in R_i and 0 otherwise, and every off-diagonal entry is 0. The Terwilliger algebra is also called the subconstituent algebra.

The Terwilliger algebra can depend on the base point x, although base points in the same orbit of the automorphism group give isomorphic algebras. Maleki and Razafimahatratra (2026) characterized when T(x) 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.

See also

Association Scheme, Bose-Mesner Algebra, Centralizer, Coherent Configuration

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

Cite this as:

Weisstein, Eric W. "Terwilliger Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TerwilligerAlgebra.html

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