## Power of a determinant with two physical applications

## Abstract

An expression for the $k$ th power of an $n\times n$ determinant in ${n}^{2}$ indeterminates $\left({z}_{ij}\right)$ is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary group $SU\left(2\right)$ , noting also the relation to the ${\text{\hspace{0.17em}}}_{3\text{\hspace{0.17em}}}{F}_{2}$ hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given.

- Authors:

- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1198451

- Grant/Contract Number:
- W-7405-ENG-36

- Resource Type:
- Published Article

- Journal Name:
- International Journal of Mathematics and Mathematical Sciences

- Additional Journal Information:
- Journal Name: International Journal of Mathematics and Mathematical Sciences Journal Volume: 22 Journal Issue: 4; Journal ID: ISSN 0161-1712

- Publisher:
- Hindawi Publishing Corporation

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

### Citation Formats

```
Louck, James D. Power of a determinant with two physical applications. Country unknown/Code not available: N. p., 1999.
Web. doi:10.1155/S0161171299227457.
```

```
Louck, James D. Power of a determinant with two physical applications. Country unknown/Code not available. doi:10.1155/S0161171299227457.
```

```
Louck, James D. Fri .
"Power of a determinant with two physical applications". Country unknown/Code not available. doi:10.1155/S0161171299227457.
```

```
@article{osti_1198451,
```

title = {Power of a determinant with two physical applications},

author = {Louck, James D.},

abstractNote = {An expression for the k th power of an n × n determinant in n 2 indeterminates ( z i j ) is given as a sum of monomials. Two applications of this expression are given: the first is the Regge generating function for the Clebsch-Gordan coefficients of the unitary group S U ( 2 ) , noting also the relation to the 3 F 2 hypergeometric series; the second is to the even powers of the Vandermonde determinant, or, equivalently, all powers of the discriminant. The second result leads to an interesting map between magic square arrays and partitions and has applications to the wave functions describing the quantum Hall effect. The generalization of this map to arbitrary square arrays of nonnegative integers, having given row and column sums, is also given.},

doi = {10.1155/S0161171299227457},

journal = {International Journal of Mathematics and Mathematical Sciences},

number = 4,

volume = 22,

place = {Country unknown/Code not available},

year = {1999},

month = {1}

}

DOI: 10.1155/S0161171299227457