# D-finite symmetric functions by Mariolys Rivas

By Mariolys Rivas

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REDUCED KRONECKER PRODUCT Partition λ λ = (m) λ = (m, 1) λ = (m, 12 ) λ = (m, l) λ = (m, l, n) 50 Reduced Kronecker coeﬃcients 1 = (l − k + m + 2) for k > m. 2 1 (l + k − m + 2) for m ≥ k. 2 λ C k,l = l − k + m + 1 for k > m. l + k − m for m ≥ k. 1 λ C k,l = (l − k + m + 1) for k > m + 1. 2 1 (l + k − m + 2) for m ≥ k − 1. 2 Algorithm. Algorithm. 1: Coeﬃcients for two-row shape Schur functions Some work has been done to characterize the coeﬃcients that appear in the Kronecker product of reduced Schur functions indexed by two-row partitions.

M Because the coeﬃcients are P-recursive, this generating function is D-ﬁnite in t. 3 in Chapter 5. 2. For the case λ = (2, 1), the previous recurrence becomes, (m − 5)(m − 1)(m − 3)am = m(m − 2)(m − 4)am−1 . Using Maple we can ﬁnd the diﬀerential equation satisﬁed by the generating function f (t), which in this case is given by, (−3t − 15)K(2,1) (t) + (3t2 + 15t) 3 dK(2,1) (t) d2 K(2,1) (t) 3 4 d K(2,1) (t) − 6t2 + (t − t ) = 0. dt dt2 dt3 CHAPTER 4. 2 26 Closure Properties As we said before, D-ﬁniteness gives information about the simplicity of the coeﬃcients of a power series.

The same holds when F is D-ﬁnite with respect to an inﬁnite number of variables. CHAPTER 4. D-FINITE FUNCTIONS 27 2. If F (x1 , x2 , . . , xn ) is D-ﬁnite in x1 , x2 , . . , xn and for each k, uk is a polynomial in the variables y1 , y2 , . . , ym , then F (u1 , u2 , . . , un ) (as long as it is well-deﬁned as a formal power series) is D-ﬁnite in y1 , y2 , . . , ym . In other words we may replace the variables x1 , . . , xn for polynomials in another ﬁnite set of variables preserving D-ﬁniteness.