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How to compute sigma in linear discriminant analysis?

The formula that I found in Elements of Statistical Learning (https://web.stanford.edu/~hastie/Papers/ESLII.pdf) on page 109 is somewhat ambiguous for me. According to it:

$\hat \sum = \sum_{k=1}^K\sum_{g_i=k}(x_i - \hat u_k)(x_i-\hat u_k)^T/(N-K)$

Now I don't understand what is $g_i$? For me it is just the $i$th categorical variable so must I compute the $\sum_{g_i=k}(x_i - \hat u_k)(x_i-\hat u_k)^T$ for every category where $x_i$ are the $x$'s given category $k$ and $u_k$ is its mean, then sum up the results and sum those for every category to get $\hat \sum$?

If it is so referring to this post (https://stackoverflow.com/questions/49331405/implement-linear-discriminant-analysis) how should one proceed with the computation of discriminant coefficients.

I tried to compute sigma using:

sigma <- cov(x[group1_index, ])+cov(x[group1_index, ])

then using the formula from the book:

enter image description here

But the matrices are not conformable because in this case the sigma is $3$ by $3$ and $x^T$ is $3$ by $25$

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