Shift Differential Operators in The Theory
نویسندگان
چکیده
we shall also denote by P [∂ x ] the differential operator P (∂ x) = P The transposition that interchanges x i and x j will be denoted s ij. It is easily shown that for any P ∈ R and 1 ≤ i < j ≤ n we have the factorization (1 − s ij)P (x) = (x i − x j) 1+2r P ij (x) I .1 with r ≥ 0, P ij (x) prime with (x i − x j) and symmetric in x i , x j. This given, a polynomial P (x) ∈ Q[X n ] is said to be " m-quasi-invariant " if and only if the difference (1 − s ij)P (x) is divisible by (x i − x j) 2m+1. The space of m-quasi-invariant polynomials in x 1 , x 2 ,. .. , x n will here and after be denoted " QI m [X n ] " or briefly " QI m ". Clearly QI m is a vector space over Q, moreover the simple identity (1 − s ij) P Q = ((1 − s ij) P)Q + (s ij P)(1 − s ij) Q I.2 shows that QI m is also a ring. Note that we have the inclusions
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