Corrigendum to ”Some new identities for Schur functions”

Frédéric Jouhet and Jiang Zeng

Institut Girard Desargues, Université Claude Bernard (Lyon 1) 43, bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France

                jouhet@euler.univ-lyon1.fr, zeng@euler.univ-lyon1.fr

Theorem 5 of [1, p. 496] is not stated correctly. For completeness, here are the correct statements.

Theorem 5. For non negative integers m and n,

∑_{λ⊆(m^{n})} f_{λ}(a, b)s_{λ}(X) = ∑_{ξ∈{±1}^{n}} β(ξ, a, b) ∏_{i} x_{i}^{m(1−ξ_{i})/2}

where the coefficient β(ξ, a, b) is equal to

\frac{a^{m}D(ξ, b)}{1 − a^{−1}b}Φ(X^{ξ}; a^{−1}, b) + \frac{b^{m}D(ξ, a)}{1 − ab^{−1}}Φ(X^{ξ}; a, b^{−1}) if |ξ|_{−1} odd,

\frac{D(ξ, ab)}{1 − ab}Φ(X^{ξ}; a, b) + \frac{(ab)^{m}D(ξ, 1)}{1 − a^{−1}b^{−1}}Φ(X^{ξ}; a^{−1}, b^{−1}) if |ξ|_{−1} even.

Indeed, the second paragraph of the page 505 of [1] should be read as follows : In the same way, we find for any even size subset Y ⊆ X that

d(Y) = −\frac{D(ξ, 1)}{1 − a^{−1}b^{−1}} Φ(X^{ξ}; a^{−1}, b^{−1})

and for any odd size subset Y ⊆ X that

a(Y) = \frac{D(ξ, b)}{1 − a^{−1}b}Φ(X^{ξ}; a^{−1}, b),
b(Y) = −\frac{D(ξ, a)}{1 − ab^{−1}} Φ(X^{ξ}; a, b^{−1}).

Consequently, the first paragraph of the page 506 of [1] should be read as follows : On the other hand, we have

β(ξ, 1, 0) = Φ(X^{ξ}; 1, 0),

β(ξ, 1, −1) = \{ \begin{matrix}Φ(X^{ξ}; 1, −1) & if m even, \\ Φ(X^{ξ}; 1, −1) ∏_{i} x_{i}^{(ξ_{i}−1)/2} & if m odd; \end{matrix}

and

β(ξ, 0, 0) = \{ \begin{matrix}0 & if |ξ|_{−1} is odd, \\ Φ(X^{ξ}; 0, 0) & otherwise. \end{matrix}

We thank Masao Ishikawa for carefully checking our theorem and pointing out the above mistake.

References

[1] F. Jouhet and J. Zeng, Some new identities for Schur functions, Adv. in Appl. Math. 27 (2001), no. 2-3, 493–509.