Representations of $\mathrm{sl}(3,\mathbb{C})$ and Catalan-like paths

Question

Background on representations of $\mathrm{sl}(3,\mathbb{C})$

In Chapter 6 of Brian C. Hall's book "Lie Groups, Lie Algebras, and Representations", he constructs the irreducible representations of $\mathrm{sl}(3,\mathbb{C})$ as follows. (I only provide the bare minimum of details.)

First note that as a vector space, $sl(3,\mathbb{C})$ has basis $X_1,X_2,X_3,Y_1,Y_2,Y_3,H_1,H_2$ where $X_i$ are strictly upper triangular, $Y_i$ are strictly lower triangular, and $H_i$ are traceless diagonal matrices. If $v$ is a weight vector (i.e. a simultaneous eigenvector of $H_1$ and $H_2$), then $X_iv$ is either zero or another weight vector. The same can be said of $Y_iv$. A highest weight vector is a weight vector that is annihilated by the $X_i$.

Now $\mathrm{sl}(3,\mathbb{C})$ has a standard action on $\mathbb{C}^3$ such that if $\{e_1,e_2,e_3\}$ is the standard basis, then $$Y_1e_1 = e_2, Y_2e_2=e_3,$$ and all other combinations give a zero. $e_1$ is the highest weight vector for this representation and the weight is $(1,0)$.

Likewise, there is the dual representation $\pi(X) := -X^{\mathrm{tr}}$, which has a basis $\{f_1,f_2,f_3\}$ for which $$Y_2f_1 = f_2, Y_1f_2=f_3,$$ and all other combinations give a zero. $f_1$ is the highest weight vector for this representation and the weight is $(0,1)$.

Hall then constructs the irreducible representation with highest weight $(m_1,m_2)$ by tensoring $m_1$ copies of the standard representation with $m_2$ copies of the dual representation. In this case, the highest weight vector is $w := e_1 \otimes \ldots \otimes e_1 \otimes f_1 \otimes \ldots \otimes f_1$, where we have $m_1$ copies of $e_1$ and $m_2$ copies of $f_1$.

Basic combinatorics of these representations

The representation is then spanned by vectors of the form $Y_{i_1}\ldots Y_{i_N} w$, where $i_k = \{1,2\}$. Moreover, the vector $Y_{i_1}\ldots Y_{i_N} w$ is either zero, or is a weight vector with weight $(m_1,m_2) - r_1(2,-1) - r_2(-1,2)$, where for $i =1,2$, $r_i$ is the number of copies of $Y_i$ in the word $Y_{i_1}\ldots Y_{i_N}$. We can see immediately (as I noted in answering this question: https://math.stackexchange.com/questions/4149260/dimension-of-weight-space-for-irreducible-representation-of-mathfraksl-left/4736165#4736165) that the dimension of the weight space $(m_1,m_2) - r_1(2,-1) - r_2(-1,2)$ is at most $\frac{(r_1+r_2)!}{r_1!r_2!}$, i.e. the number of different words involving $r_1$ copies of $Y_1$ and $r_2$ copies of $Y_2$.

We can do better than this crude inequality however. As a warm up, let us consider the representation with highest weight $(m_1,0)$. This representation is spanned by vectors of the form $e_1^{m_1-r-s}e_2^{r}e_3^s$, where I'm using this as shorthand for the symmetric product of the sum of all tensor words involving $m_1-r-2$ copies of $e_1$, $r$ copies of $e_2$ and $s$ copies of $e_3$. Applying a word $Y_{i_1}\ldots Y_{i_N}$ to the highest weight vector $w=e_1^{\otimes m_1}$, we see that this vector is nonzero only if every initial subword has at least as many $Y_1$s as $Y_2$s. In short, we can think of the word being associated with a walk (with $Y_1$s denoting ups, and $Y_2$s denoting downs) that always stays above the horizontal axis.

Turning to the general case of the representation with highest weight $(m_1,m_2)$, since the product of a Lie algebra representation works via $(\pi_1 \otimes \pi_2)(X) := \pi_1(X) \otimes I + I \otimes \pi_2(X)$, it follows that we can view each copy of $Y_i$ as either acting on $e_1^{\otimes m_1}$ or on $f_1^{\otimes m_2}$. In particular, with $w = e_1^{\otimes m_1} \otimes f_1^{\otimes m_2}$ we have $$Y_{i_1}\ldots Y_{i_N}w = \sum_{ A \subset [N] } \prod_{a \in A} Y_{i_a} e_1^{\otimes m_1} \prod_{a \in [N]-A} Y_{i_a} f_1^{\otimes m_2},$$ where the products are ordered.

The term $Y_{i_a} e_1^{\otimes m_1}$ is nonzero only if every initial subword of $(Y_{i_a} : a \in A )$ contains at least as many $Y_1$s as $Y_2$s, and conversely, $ Y_{i_a} f_1^{\otimes m_2} $ is only nonzero if every initial subword of $(Y_{i_a} : a \in [N]-A)$ contains at least as many $Y_2$s as $Y_1s$.

Equivalently,

1. We associate the word $Y_{i_1}\ldots Y_{i_N}$ with a (black) up/down walk. Where the first step is up if $i_N=1$, and down if $i_N=2$ etc.
2. We sum over all red/blue colourings of the $N$ steps of this walk. (The steps labelled red correspond to the set $A$ above.) This creates a red walk and a blue walk. The red/blue walks have up/down/flat steps, where the red/blue walk at time $t$ is the sum over all red/blue steps so far. The term in the colouring is nonzero only if the red walk stays above the horizontal axis and the blue walk stays below the horizontal axis.

So to reiterate, the weight space associated with $(m_1,m_2) - r_1(2,-1) - r_2(-1,2)$ is associated with all up/down walks with $r_1+r_2$ steps, $r_1$ of which are up. Each of these walks has a red/blue colouring, which gives a nonzero term if all initial sums of the red (resp. blue) steps stays above (resp. below) the horizontal axis.

Question:

In principle, there is a combinatorial calculation that can now be done to obtain a formula for the dimension of the weight spaces. In particular, one should be able to obtain the fact that the total dimension of the representation is $$\frac{1}{2}(m_1+1)(m_2+1)(m_1+m_2+2).$$ Does this kind of combinatorial approach to the dimensions of weights spaces appear in the literature anywhere? How does this relate to the action of the Weyl group on weight space? Presumably the invariance of dimensions of weight spaces under the action of the Weyl group relates to some sort of symmetry property of decompositions of up/down walks. Can the same be done for $\mathrm{sl}(n,\mathbb{C})$, or even more general Lie algebras (perhaps with the aid of Verma modules)? Is there a Brownian analogue of the above discussion as $m_1,m_2$ are sent to infinity under a suitable scaling limit?

Thank you in advance for any responses!

Literature and discussion I was able to find

1. Stackexchange question about Catalan numbers and $SU(2)$
2. Stanley's many representations of the Catalan numbers mention representation-theoretic interpretations, but nothing exactly related to this discussion.
3. Wildberger - Quarks, Diamonds and representations of $\mathrm{sl}(3,\mathbb{C})$
4. Littelmann, Peter, Paths and root operators in representation theory, Ann. Math. (2) 142, No. 3, 499-525 (1995). ZBL0858.17023.
5. Biane, Philippe; Bougerol, Philippe; O’Connell, Neil, Littelmann paths and Brownian paths, Duke Math. J. 130, No. 1, 127-167 (2005). ZBL1161.60330.