Talarico, Marco2024-12-162024-12-162024-12-16http://hdl.handle.net/10393/49979https://doi.org/10.20381/ruor-30783In this thesis we construct a generalization of the higher Specht polynomials to the Hecke algebra 𝓗_𝑞(𝑆_𝑛). These polynomials form a basis of the coinvariant algebra 𝕮 with respect to the action of 𝑆_𝑛, and they will decompose 𝕮 into irreducible representations of the Hecke algebra. These irreducible representations are 𝑞-Specht modules 𝑆_λ^𝑞. In this construction, if we consider 𝑞 = 1 then we obtain the original higher Specht polynomials for 𝑆_𝑛. We will also introduce a generalization of the divided difference and Demazure operators in the setting of the ring of Laurent polynomials 𝔏. We will construct a coinvariant algebra for the action of the hyperoctahedral group 𝑊_𝑛 on 𝔏. From these operators, we will be able to find a faithful representation of the Hecke algebra 𝓗_{𝑞,𝑝}(𝑊_𝑛) over 𝔏.enRepresentation TheoryCombinatoricsHecke AlgebrasThesis