Correction to ‘Toeplitz Minors’

Daniel Bump and Persi Diaconis

July 14, 2003

Barry Simon pointed out a point requiring clarification in the proof of Theorem 4. After the displayed formula

e^{nc_{0}} ∫_{U(n)} ∏_{k=1}^{∞} ∑_{a_{k}=0}^{∞} \frac{(c_{k} tr(g^{k}))^{α_{k}}}{α_{k}!} ∑_{a_{k}=0}^{∞} \frac{(c_{−k}\overline{tr(g^{k}}))^{β_{k}}}{β_{k}!} dg,

it is asserted that “expanding this and invoking (1.11) and (1.12), the only terms which survive have α_{k} = β_{k}.”

This is an oversimplification. It is true that if either ∑ kα_{k} ⩽ n or ∑ kβ_{k} ⩽ n then the term is killed by (1.11). However if ∑ kα_{k} > n and ∑ kβ_{k} > n all we can immediately assert is that ∑ kα_{k} = ∑ kβ_{k}. (For this, integration over the center is sufficient to kill the term if ∑ kα_{k} ≠ ∑ kβ_{k}.)

To correct this point, we will not use (1.12) but just the fact that the characteristic map R_{m} ⟶ Λ_{m} is a subisometry (first displayed formula after (1.8)).

Expanding and collecting terms with ∑ kα_{k} = ∑ kβ_{k} = m, we obtain

e^{nc_{0}} ∑_{m=0}^{∞} C(m, n),

where

C(m, n) = ⟨ ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} (∏_{k} \frac{c_{k}^{α_{k}}}{α_{k}!} ) p_{λ_{α}}, ∑_{\begin{matrix}(β_{k}) \\ ∑ kβ_{k} = m \end{matrix}} (∏_{k} \frac{\overline{c_{−k}}^{β_{k}}}{β_{k}!} ) p_{λ_{β}}⟩ .

Here as in the paper λ_{α} is the partition with α_{k} parts equal to k. When m ⩽ n, we may simply expand the inner product and use (1.11) to simplify the result as in the paper. When m ⩽ n we can discard the terms with α_{k} ≠ β_{k} and write

C(m, n) = ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} (∏_{k} \frac{c_{k}^{α_{k}}}{α_{k}!} ) (∏_{k} \frac{c_{−k}^{α_{k}}}{α_{k}!} ) ⟨p_{λ_{α}}, p_{λ_{α}}⟩ = C(m)

where

C(m) = ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} ∏_{k} \frac{(kc_{k}c_{−k})^{α_{k}}}{α_{k}!} .

This much is as in the paper. If n > m we will show that

|C(m, n)| ⩽ |D(m)|,

where the series

∑_{m} |D(m)| < ∞.

This is clearly sufficient for the purposes of the theorem.

First consider the case where c_{k} = \overline{c_{−k}}. In this case, we can take D(m) = C(m).

|C(m, n)| = ∥ ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} (∏_{k} \frac{c_{k}^{α_{k}}}{α_{k}!} ) p_{λ_{α}} ∥^{2}

which is dominated by the corresponding inner product in R_{m} because the characteristic map is a subisometry. Thus (with c_{k} = \overline{c_{−k}})

|C(m, n)| ⩽ ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} ∑_{\begin{matrix}(β_{k}) \\ ∑ kβ_{k} = m \end{matrix}} (∏_{k} \frac{c_{k}^{α_{k}}}{α_{k}!} ) (∏_{k} \frac{c̄_{k}^{β_{k}}}{β_{k}!} ) ⟨z_{λ_{α}}f_{λ_{α}}, z_{λ_{β}}f_{λ_{β}}⟩

where now the inner product is in R_{m}. The right-hand side equals C(m) as before. Note that because this is an inner product on R_{m} the f_{λ_{α}} are orthogonal, even though (if m > n) their images under the characteristic map are not orthogonal.

If c_{k} and \overline{c_{−k}} are not assumed equal, we may use the Cauchy-Schwarz inequality and write

|C(m, n)| ⩽ ∥ ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} (∏_{k} \frac{c_{k}^{α_{k}}}{α_{k}!} ) p_{λ_{α}} ∥ · ∥ ∑_{\begin{matrix}(α_{k}) \\ ∑ kα_{k} = m \end{matrix}} (∏_{k} \frac{c_{−k}^{α_{k}}}{α_{k}!} ) p_{λ_{α}} ∥ .

Each norm is dominated by the corresponding norm in R_{k} and proceeding as before, we obtain the same bound with c_{k} replaced by max(|c_{k}|, |c_{−k}|).