We give the presentations of the symmetric group and a generalization
known as the Hecke algebra.
We introduce standard Young tableaux, permutations, and a bijection
between them, called the Robinson-Schensted correspondence. These objects
are crucial to define the Kazhdan-Lusztig basis, which is instrumental in
the representation theory of the Hecke algebra.
Now we focus our attention on the polynomial ring on n^2 variables, and a
generalization called the quantum polynomial ring. We develop an analog of
the previous Kazhdan-Lusztig basis in this polynomial ring, called the
Kazhdan-Lusztig immanants. We state vanishing results for these
Kazhdan-Lusztig immanants. Applying these results we find a new
construction of the Kazhdan-Lusztig representations of the Hecke algebra.
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