Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)
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The symmetric Macdonald difference operator of type A1

Definition

Let V={f∈K[z,z−1]:f(z)=f(z−1)} be the Weyl-invariant Laurent polynomials. Define the type-A1 symmetric Macdonald difference operator on V by

(Df)(z)=tz2−t−1z2−1f(qz)+t−1z2−tz2−1f(q−1z).

The formula is interpreted as the sum of the two rational terms; The A1 Macdonald operator is triangular on Weyl orbit sums ↗ proves that for every f∈V the sum is a Weyl-invariant Laurent polynomial.

Remarks

The symmetric DAHA translation restricts to the A1 Macdonald operator identifies this operator as the restriction of the symmetric DAHA translation operator.

Depends on

Used by

Dependency tree · two levels

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Sources