Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a $r$-part partition of integer $N$ ($\lambda\vdash N$), i.e., $$\sum_{i=1}^r{\lambda_i}=N,$$ such that $\lambda_i\leq\lambda_j$ for $r\geq j>i\geq1$. Thus, $\lambda$ defines a Young diagram.
In addition, let $\mathbf{y}^{(k)}_{\lambda}$ be the $k$th standard Young tableau, SYT, of shape $\lambda$, i.e., the array of numbers $1\leq y_{i,j}\leq N$ that fulfills
- $y_{i,j}>y_{i-1,j}$
- $y_{i,j}>y_{i,j-1}$
for indices $1\leq i \leq r$ and $1 \leq j \leq\lambda_i$. The number of possible SYT of shape $\lambda$ is $\dim S^\lambda$, the dimension of the Specht module.
Moreover, let us define the permutation operator $\hat{\pi}=(\hat{\pi}_1,\hat{\pi}_2,\ldots,\hat{\pi}_r)$, such that $$\mathbf{w}=\hat{\pi}\mathbf{y},$$ where $\textrm{row}_i[\mathbf{w}]=\hat{\pi}_i\textrm{row}_i[\mathbf{y}]$, with $\textrm{row}_i[\mathbf{w}]$ the $i$th row of array $\mathbf{w}$, and $\hat{\pi}_i\in S_{\lambda_i}$.
Finally, let $$\textrm{Spe}\left(\mathbf{w}\right)=\prod_{i=1}^{\lambda_1}V\left(\textrm{col}_i[\mathbf{w}]\right)$$ be the Specht polynomial for the (not-necessarily standard) Young tableau $\mathbf w$, where $\textrm{col}_i[\mathbf{w}]$ is the $i$th column of array $\mathbf{w}$, and $$V\left(a_1,a_2,\ldots,a_n\right)=\prod_{1\leq j<k\leq n}(x_{a_j}-x_{a_k})$$ is the Vandermonde determinant of set $\{a_1,a_2,\ldots,a_n\}$.
After playing around with Gram-Schmidt orthogonalization, I have formulated the following conjecture:
The set of polynomials $$\left\{p_\lambda^{(k)}\left(x\right):1\leq k\leq \dim S^\lambda\right\},$$ where $x=\{x_i:1\leq i\leq r\}$, $$p_\lambda^{(k)}\left(x\right)=\sum_{c}\textrm{Spe}\left(\mathbf{w}_\lambda^{(c,k)}\right),$$ and $\mathbf{w}_\lambda^{(c,k)}=\hat{\pi}^{(c)}\mathbf{y}_\lambda^{(k)}$, with the index $c$ running over all possible permutation operators that produce Young tableaux with the constraint $$w_{i,j}>w_{i,j-1},$$ forms an orthogonal basis in the domain $-1/2\leq x_{1\leq i\leq r} \leq 1/2$, i.e., $$\int_{-1/2}^{1/2} p_\lambda^{(k)}(x)p_\lambda^{(k')}(x)\textrm{d}^{r}x\propto\delta_{k,k'}.$$
My question is, how can I go about a general proof that does not involve checking case by case and performing all inner products one by one?