show/hide this revision's text 2 Corrected index of Spin group from 2k-1 to 2k+1

I don't know if this qualifies as less trivial than your examples since it relies on the same identity that your example (1) does, but here goes:

Define $f_k(x_1,\ldots,x_k) = \frac{(1+x_1)(x_{k-1}+x_k^2)}{\prod_{i=1}^kx_i}\prod_{j=2}^{k-1}(x_{j-1}+x_j)$.

Then the constant term of $f_k^{p-1}$ is $\binom{p-1}{\frac{p-1}{2}}^k \equiv \pm 1 \mod p$ (note that your congruence in (1) should be $\pm 1$ as well). Furthermore, if one exchanges any number of the plus signs with minus signs, the constant term remains the same up to possibly a factor of -1; this does not affect the congruence $\mod p$ however. In light of this, your example (1) is the special case where the plus sign in $f_1$ is replaced by a minus sign.

The reason this particular sequence of $f_k$'s came to mind is that relative to certain coordinates on a maximal torus, via the Weyl Character Formula $f_k$ represents the character of the spinor representation of $Spin(2k-1)$. Spin(2k+1)$. Extracting the constant term of $f_k^n$ is then equivalent to counting the multiplicity of the zero weight in the $n^{th}$ tensor power of the spinor representation.

I have been toying around with the Laurent polynomials corresponding to other representations of compact groups, but have not yet found any others that (provably) satisfy your criterion.
show/hide this revision's text 1

I don't know if this qualifies as less trivial than your examples since it relies on the same identity that your example (1) does, but here goes:

Define $f_k(x_1,\ldots,x_k) = \frac{(1+x_1)(x_{k-1}+x_k^2)}{\prod_{i=1}^kx_i}\prod_{j=2}^{k-1}(x_{j-1}+x_j)$.

Then the constant term of $f_k^{p-1}$ is $\binom{p-1}{\frac{p-1}{2}}^k \equiv \pm 1 \mod p$ (note that your congruence in (1) should be $\pm 1$ as well). Furthermore, if one exchanges any number of the plus signs with minus signs, the constant term remains the same up to possibly a factor of -1; this does not affect the congruence $\mod p$ however. In light of this, your example (1) is the special case where the plus sign in $f_1$ is replaced by a minus sign.

The reason this particular sequence of $f_k$'s came to mind is that relative to certain coordinates on a maximal torus, via the Weyl Character Formula $f_k$ represents the character of the spinor representation of $Spin(2k-1)$. Extracting the constant term of $f_k^n$ is then equivalent to counting the multiplicity of the zero weight in the $n^{th}$ tensor power of the spinor representation.

I have been toying around with the Laurent polynomials corresponding to other representations of compact groups, but have not yet found any others that (provably) satisfy your criterion.