Asymptotic Behavior of Wronskian Polynomials that are Factorized via p -cores and p -quotients
- PDF / 1,300,368 Bytes
- 27 Pages / 439.642 x 666.49 pts Page_size
- 17 Downloads / 158 Views
Asymptotic Behavior of Wronskian Polynomials that are Factorized via p -cores and p -quotients Niels Bonneux1 Received: 8 May 2020 / Accepted: 31 August 2020 / © Springer Nature B.V. 2020
Abstract In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of p-cores and p-quotients. We obtain the asymptotic behavior for these polynomials when the p-quotient is fixed while the size of the p-core grows to infinity. For this purpose, we associate the p-core with its characteristic vector and let all entries of this vector simultaneously tend to infinity. This result generalizes the Wronskian Hermite setting which is recovered when p = 2. Keywords Asymptotics · Cores and quotients · Partitions · Wronskians
1 Introduction In this paper we study the asymptotic behavior of the Wronskian polynomials i−1 d det dx q i−1 nj Wr[qn1 , qn2 , . . . , qnr ] 1≤i,j ≤r = (1.1) qλ = (nλ ) (nj − ni ) i n2 > · · · > nr > 0. We indicate the degree vector in the associated Maya diagram Mλ which is defined as Mλ = {n ∈ Z | n < 0} ∪ {ni | 1 ≤ i ≤ r} ⊂ Z.
(2.1)
This diagram can be visualized by a doubly-infinite sequence of consecutive boxes that are either empty or are filled with a bullet. The boxes are labeled by the integers and the nth box is filled precisely when n ∈ Mλ . Furthermore, a vertical line is placed between the boxes labeled with −1 and 0; subsequently we can omit the labels. We may shift the vertical line such that the sequence of filled and empty boxes remains unchanged, but the labeling differs. We call such Maya diagrams equivalent to Mλ and denote them by Mλ + t where the integer t indicates the shift, that is, m ∈ Mλ if and only if m + t ∈ Mλ + t. There is a unique shift such that the number of empty boxes to the left of the vertical line equals the number of filled boxes to the right of λ to denote this specific Maya diagram. In mathematical physics, see it. We write M for example [13], particles represent filled boxes while empty boxes are called holes. Moreover, the number of filled boxes to the right minus the number of empty boxes λ to the left of the vertical line is usually called the charge of the diagram, and so M has charge zero. Using this definition, we can create a bijection which maps Maya diagrams to their charge and associated partition so that equivalent Maya diagrams are mapped to the same partition but different charges. Example 2.1 Let λ = (8, 8, 6, 6, 2, 2, 1) so that nλ = (14, 13, 10, 9, 4, 3, 1). The integers in the latter sequence are indicated to the right of the vertical line in the first Maya diagram of Fig. 1. If we move the vertical line 7 steps to the right, then there λ = Mλ − 7. are 4 empty boxes to the left of it and 4 filled boxes to the right. So M λ reveals the Young diagram (in Russian style) of the partition The Maya diagram M as shown in Fig. 2. Each filled dot corresponds to a downwards step whereas the empty ones give rise to an upwards step. With each partition λ we can associate its p-core and its p-qu
Data Loading...
