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I would like to ask a very concrete question. First I recap what I know

  1. One can label a representation $\mathcal{R}$ of $\mathrm{U}(N)$ with a sequence of ordered integers (positive, negative or null): $$\rho_1 \geq \rho_2 \geq \dots \geq \rho_N \;. $$
  2. Characters of $\mathrm{U}(N)$ are $$\chi_{\rho}(x_1, \dots, x_N) = \frac{a_{(\rho_1+N-1, \rho_2+N-2, \dots, \rho_N )}(x_1,\dots,x_N)}{a_{(N-1, N-2, \dots, 0 )}(x_1,\dots,x_N)} \;, $$ where $$a_{(k_1, \dots k_N)}(x_1,\dots x_N) = \det \left(x_i^{k_j} \right)_{i,j=1}^N \;.$$
  3. In particular, when all $\rho_i \geq 0$ we have that the character coincides with the corresponding Schur polynomial $$χ_\rho(x_1,\dots,x_N) = s_\rho(x_1,\dots,x_N)\;. $$
  4. We can introduce power sums associated to a partition $\lambda = (\lambda_1, \dots, \lambda_\ell)$. $$p_k(x_1,\dots,x_N) = \sum_{i=1}^Nx_i^k \;, \qquad p_\lambda(x_1,\dots,x_N)= \prod_{j=1}^\ell p_{\lambda_\ell(x_1,\dots,x_N)} \;. $$ Then it is possible to express Schur polynomial in terms of these power sums and viceversa by means of $$ s_\lambda(x_1,\dots,x_N) = \sum_\eta z_\eta^{-1} \chi_\lambda^\eta p_\eta(x_1,\dots,x_N)\;, \tag{1}$$ $$ p_\lambda(x_1,\dots,x_N) = \sum_\eta \chi_\eta^\lambda s_\eta(x_1,\dots,x_N) \;, \tag{2}$$ where $z_\lambda = \prod_n n^{a_n}a_n!$ where $n$ is a number appearing $a_n$ times in the partition $\lambda$, and $\chi_\lambda^\eta$ is the coefficient of $x_1^{\lambda_1+m-1}x_2^{\lambda_2+m-2}\cdot\dots\cdot x_m^{\lambda_m}$ in the expansion of $\prod_{1\leq i < j \leq m}(x_i-x_j)p_\eta(x_1,\dots,x_m)$. In particular, formula $(1)$ is dubbed "Frobenius formula".

I noticed, however that even if some of $\rho_i$ are negative it is possible to expand $χ_\rho$, which, in this case is a rational function, in terms of power sums, in this case including also negative value of $k$. For example $$χ_{(1,-1)}(x_1,x_2) = 2 + x_1x_2^{-1} + x_1^{-1}x_2 = -\frac{1}{2}p_{\emptyset}(x_1,x_2) + p_{(1,-1)}(x_1,x_2) \\= -\frac{1}{2}p_0(x_1,x_2) + p_1(x_1,x_2)p_{-1}(x_1,x_2) \;,$$

QUESTION

Is there a way to generalize eqs. $(1)$ and $(2)$ for a generic character of $\mathrm{U}(N)$ (not only those corresponding to $\rho_i$ all positive), that is to write any of these characters as combination of power sums and viceversa?

If yes what is the concrete experssion for $\chi_\lambda^\eta$ and $z_\lambda$?

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