# Construct Hecke algebra as a ring

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### Yingdong Li

May 25, 2022, 12:57:57 PM5/25/22
to sage-nt
Dear all,

Motivated by problems about modular forms, we want to find the ring structure of Hecke algebra. Therefore, I have written some codes in Sage to compute the finite-dimensional algebra by a list of commuting matrices and I want to contribute it to Sage. Here is the idea of my codes.

1. We can construct the algebra as a quotient of a polynomial ring(by using the homomorphism which sends each x_i to t_i, where t_1,...,t_n is the n matrices generate the algebra), we can also get the basis by doing this.

2. With the basis of the algebra, we can also compute the multiplication table then use the finite-dimensional algebra command in Sage to get a description to this algebra.

Once we have done with these things above, we can get the ring structure of the algebra. This is very useful in dealing with some problems about modular forms since we can further study the prime ideals or maximal ideals of Hecke algebra by using its ring structure.

I'm an undergraduate student and this is part of my research project. I was wondering how I can contribute the codes to Sage. Could anyone give me some help me with this(since I'm not so familiar about the Sage trac and I'm not sure where I can share my codes)? Thanks in advance!

Moreover, if you have some questions or comments on this, we can discuss about it here.

Here's a link of my code in GiHub(see the code called "Finite generated algebra as a ring")

Best wishes,
Li Yingdong