# Peak algebra

In mathematics, the **peak algebra** is a (non-unital) subalgebra of the group algebra of the symmetric group *S*_{n}, studied by Nyman (2003). It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2,...,*n*} is an index *i* such that σ(*i*–1)<σ(*i*)>σ(*i*+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all *n* has a natural structure of a Hopf algebra.

## References

- Nyman, Kathryn L. (2003), "The peak algebra of the symmetric group",
*J. Algebraic Combin.*,**17**(3): 309–322, doi:10.1023/A:1025000905826, MR 2001673

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