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I am not aware of that many identities that involve several Littlewood–Richardson coefficients. One recent identity, is a generating function as sum of squares of LR-coefficients, due to Harris and Willenbring in Sums of squares of Littlewood–Richardson coefficients and $\operatorname{GL}_n$-harmonic polynomials. This identity is really interesting.

There is also a relation between two sums of LR-coefficients, due to Coquereaux and Zuber, but I find this less interesting for this question.

So my question is, are there some interesting relations involving (potentially weighted) sums of LR-coefficients?

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    $\begingroup$ You can take the definition of the LR coefficients $s_{\mu}(x_1,\ldots) s_{\nu}(x_1,\ldots) = \sum_{\lambda} c^{\lambda}_{\mu, \nu} s_{\lambda}(x_1,\ldots)$ and do any of the several nice specializations of Schur functions to get such an identity. $\endgroup$
    – Sam Hopkins
    Commented Jul 30, 2022 at 19:37
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    $\begingroup$ There are other identities involving LR coefficients which follow from exceptional equalities among skew Schur functions (see e.g. arxiv.org/abs/math/0602634) or sums of products of pairs of Schur functions (see e.g. arxiv.org/abs/math/0004113), but these are of a different flavor. $\endgroup$
    – Sam Hopkins
    Commented Jul 30, 2022 at 19:47
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    $\begingroup$ For another example: arxiv.org/abs/2008.06128 $\endgroup$
    – darij grinberg
    Commented Jul 30, 2022 at 21:01
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    $\begingroup$ For a mild generalization of the Harris-Willenbring identity, see Problem 92(d) at math.mit.edu/~rstan/ec/ch7supp.pdf. Problem 91 also has some identities involving $c^\lambda_{\mu\nu}$ together with $f^\lambda$, $f^\mu$, and $f^\nu$. $\endgroup$
    – Richard Stanley
    Commented Jul 31, 2022 at 0:55
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    $\begingroup$ The OP was asked in relation to some new results which have been finally made available here: arxiv.org/abs/2306.11115. In this paper, a relation involving the Jack Littlewood-Richardson coefficients of the following form is shown $\sum_{\gamma \supset \mu \cup \nu} c_{\mu\nu}^{\gamma}(\alpha) f_{\gamma,\mu,\nu}(u) = g_{\mu,\nu}(u)$, where $f_{\gamma,\mu,\nu}(u)$ and $g_{\mu,\nu}(u)$ are explict rational functions in a variable $u$. $\endgroup$
    – Ryan Mickler
    Commented Jun 21, 2023 at 2:20

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We looked into this with Greta and Damir in On the largest Kronecker and Littlewood–Richardson coefficients. Unfortunately, other than the Harris–Willenbring identity we didn't find much. For our bounds we even had to invent a new (not too difficult) LR-identity which follows from the skew Cauchy identity (it's Lemma 5.1 in the paper): $$ \sum_{\lambda\vdash n} \left( c^{\lambda}_{\mu\nu}\right)^2 \, = \, \sum_{\alpha, \beta, \gamma, \delta} c^{\mu}_{\alpha\gamma} c^{\mu}_{\alpha \delta} c^{\nu}_{\beta\gamma} c^{\nu}_{\beta \delta}. $$

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