Euler’s triangle and the decomposition of tensor powers of the adjoint 𝔰𝔩(2)-module
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By considering a relation between Euler’s trinomial problem and the problem of decomposing tensor powers of the adjoint 𝔰𝔩(2)-module I derive some new results for both problems, as announced in arXiv:1902.08065.
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In 1765, Euler  investigated the coefficients of trinomial
For central trinomial coefficients he found the generating function and a two-term recurrence relation. For a discussion of properties of the , see .
Let us change variable x by exp(iθ) and rewrite the left-hand side of (1.1) as
Note that X is the character χ1 of the adjoint 𝔰𝔩(2)-module. In what follows, Xn denotes both the representation with character Xn, and the corresponding module.
So, Euler’s problem is equivalent to the problem of multiplicities of weights in the representation with character Xn. I also consider, related to the above, the problem of decomposing Xn into irreducible 𝔰𝔩(2)-modules.
2. Euler’s triangle
It is evident that . So, it suffices to consider only quantities for k ≥ 0. It is convenient to arrange these coefficients in a triangle. I give here the table of these numbers till n = 10:
Eq. (1.1) immediately implies the three-term recurrence relation
Introduce the generating function F(t) for the central trinomial coefficients:
Theorem 2.1 (Euler 1765).
The following statements hold.
The generating function F(t) has the form(2.3)
For the an, the following two-term recurrence relation takes place(2.4)
We give here a very short proof of item 1); it is different from Euler’s.
Evaluating this integral we obtain formula (2.3).
Item 2) is a special subcase of the following more general statement.
For the , there is the following two-term recurrence relation
This implies formula (2.5).
For the , there are the following two-term recurrence relations:
From the identitywe obtain relation (2.6). Combining this relation with (2.2), we obtain relations (2.7)–(2.9).
Note that eq. (2.2) implies
3. Decomposition of Xn into irreducible representations
This problem is equivalent to expanding Xn in terms of characters of 𝔰𝔩(2)-modules:
These characters are well known (see, for example, ):
They are orthogonaland we have
This implies the basic relationand a three-term recurrence relation similar to relation (2.2) as well as the following relations
The triangle for the numbers analogous to the triangle (2.1) is as follows.
The generating function is of the form
Taking into account the identitywe reduce the proof to the proof for F(t). We also have the recurrence relation which follows from eq. (2.4) and the equality .
There is a four-term recurrence relationwhere
There is the following three-term recurrence relation
This follows from eq. (2.7) and the relation .
I am thankful to D. Leites who improved my English in this letter.
Cite this article
TY - JOUR AU - Askold M. Perelomov PY - 2019 DA - 2019/10 TI - Euler’s triangle and the decomposition of tensor powers of the adjoint 𝔰𝔩(2)-module JO - Journal of Nonlinear Mathematical Physics SP - 1 EP - 6 VL - 27 IS - 1 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1684001 DO - https://doi.org/10.1080/14029251.2020.1684001 ID - Perelomov2019 ER -