**Proof.** Let $a$ be either 1 or 2. Now let describe the connection between the word corresponding to $\bar{\lambda}\bar{\lambda} \in C_{2n+a}^{dd}$ and $\bar{\lambda}$. Set $\phi_{2n+a}(\bar{\lambda}\bar{\lambda}) = (m_0, m_1, \dots, m_n, m_{n+1}, m_{n+2}, \dots, m_{2n+a-1})$ (see Definition 6.8) and $\psi(\bar{\lambda}\bar{\lambda}) = (c_k)_{k \in \mathbb{Z}}$ (see Definition 6.1), then by definition of $\phi_{2n+a}$, $(2n+a)m_i + i = \max\{(k+1)g + i \mid c_{kg+i} = 0\}$ for any $0 \le i \le 2n+a-1$. Take $1 \le i \le n$. If $\max(m_{2n+a-i}, m_i) > 0$, then Lemma 6.4 guarantees that there exists $k \in \{1, \dots, \ell\}$ such that $\bar{\lambda}_k = \max((2n+a)(m_{2n+a-i}-1) + 2n+a-i, (2n+a)(m_i-1)+i)$. Moreover if $m_i = 0$, by Lemma 6.4, this is then equivalent to

$$ \{\bar{\lambda}_k, k \in \{1, \dots, \ell\} \mid \bar{\lambda}_k \equiv \pm i \pmod{2n+a}\} = \emptyset. $$

Therefore we have the set equality:

$$ \{\bar{\lambda}_1, \dots, \bar{\lambda}_\ell\} = \cup_{i=1}^{2n+a-1} \{(2n+a)k + i, 0 \le k \le m_i - 1 \mid m_i > 0\}. $$

<figure><img src="image_13.png" /></figure>

As already mentioned, when $\bar{\lambda}$ belongs to $C_n^{d,\mathrm{tr}}$, its doubled version $\bar{\lambda}\bar{\lambda}$ does not belong to $C_n^{dd,\mathrm{tr}}$. Nevertheless, the core and the quotient of $\bar{\lambda}\bar{\lambda}$ take a very particular form as explained in the following proposition which can be deduced from Lemma 6.7.

**Proposition 6.15.** Let $n$ be an integer greater or equal to 2. Take $\bar{\lambda} \in \mathcal{D}$ and set $\Phi_n((\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}) = (\omega, \nu^{(0)}, \dots, \nu^{(n-1)})$ (see Definition 6.5). Then $\bar{\lambda}$ belongs to $C_n^{d,\mathrm{tr},r}$ if and only if:

* $\nu^{(j)} = \emptyset$ for any $0 \le j \le n-2$,

* let $m = \max(\{1\} \cup \{(n+\bar{\lambda}_i)/n \mid \bar{\lambda}_i \equiv 0 \pmod{n}\})$. Then $\nu^{(n-1)}$ is the rectangle $(m-1) \times m$ partition.

Moreover, in the odd case, setting $\phi_{2n-1}(\omega) = (m_0, \dots, m_{n-2}, -m_{n-2}, \dots, -m_0, 0)$ one gets

$$ (31) \qquad |\bar{\lambda}| = \frac{2n-1}{2} \left( m^2 + \sum_{i=0}^{n-2} m_i^2 \right) + \left( -\frac{2n-1}{2} m + \sum_{i=0}^{n-2} (i-n+\frac{1}{2})m_i \right). $$

In the even case, by setting $\phi_{2n}(\omega) = (m_0, \dots, m_{n-2}, 0, -m_{n-2}, \dots, -m_0, 0)$ one gets

$$ (32) \qquad |\bar{\lambda}| = n \left( m^2 + \sum_{i=0}^{n-2} m_i^2 \right) + \left( -nm + \sum_{i=0}^{n-2} (i-n+1)n_i \right). $$

**Proof.** Consider $\bar{\lambda} \in C_n^{d,\mathrm{tr},r}$, the Littlewood decomposition $\Phi_n((\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}) = (\omega, \nu^{(0)}, \dots, \nu^{(n-1)})$ (see Definition 6.5) and $\psi(\bar{\lambda}\bar{\lambda})^{\mathrm{tr}} = (c_k)_{k \in \mathbb{Z}}$ its corresponding word (see Definition 6.1). Since $n/2 \notin \bar{\lambda}$, the main diagonal $\Delta$ of $(\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}$ has no box of hook length $n$. If there exists $i \in \{0, \dots, n-2\}$ such that $\nu^{(i)} \ne \emptyset$, there exists at least one box in $\nu^{(k)}$ whose hook length is equal to 1. By Lemma 6.2 and by property (DD'4) from Lemma 6.7, there exists $m \in \mathbb{Z}$ such that $c_{mn+i} = 1$ and $c_{(m+1)n+i} = 0$. Note that the hook length of the box $(mn+i, (m+1)n+i)$ is equal to $n$. If $i \ne -i-2$ (mod $n$), that is if $i \ne n/2$, by (25), one gets that $c_{-(m+1)n-i-2} = 1$ and $c_{-mn-i-2} = 0$. Moreover one of the two boxes corresponding to the pairs of indices $(mn+i, (m+1)n+i)$ and $-(m+1)n-i-2, -mn-i-2)$ is strictly above the main diagonal $\Delta$, which contradicts the fact that $\bar{\lambda} \in C_n^{d,\mathrm{tr},r}$. If $i = n/2$, which implies that $n$ is even in this case, using the same arguments, $\bar{\lambda} \in C_n^{d,\mathrm{tr},r}$ implies that there is no hook of length $n$ apart maybe on the diagonal $\Delta$. Since that $n/2 \notin \bar{\lambda}$, we have that $\nu^{(n/2)} = \emptyset$. Therefore $\bar{\lambda} \in C_n^{d,\mathrm{tr},r}$ implies that $\nu^{(i)} = \emptyset$ for any $0 \le j \le n-2$.

Now we prove the second part of the statement. Let $d$ be the size of the Durfee square of $\nu^{(n-1)}$. Using the same arguments as above, if $\nu^{(n-1)}$ has a hook length equal to 1 in any row but the $d+1$-th row, the properties of symmetry along the main diagonal and the property (DD'4) from Lemma 6.7 imply that $\nu^{(n-1)}$ contains a unique box whose hook length is equal to 1. Therefore $\nu^{(n-1)}$ is a rectangle of size $m \times (m-1)$ with $m = \max(\{1\} \cup \{(n+\bar{\lambda}_i)/n \mid \bar{\lambda}_i \equiv 0 \pmod{n}\})$ or equivalently $m = \max(k+1 \mid c_{kn+n-1} = 0)$. Note that the word to $\nu^{(n-1)}$ is:

$$ \psi(\nu^{(n-1)}) = \dots 0 \underbrace{1 \dots 1}_{m-1} \underbrace{0 \mid 0 \dots 0}_{m} 1 \dots $$

Equalities (31) and (32) are derived using (30) and Lemma 6.7 (DD'3).

<figure><img src="image_14.png" /></figure>

Using the same arguments as for Proposition 6.15, we obtain similarly for $C_n^{d,\mathrm{tr}}$.

**Proposition 6.16.** Let $n$ be an integer greater or equal to 2. Take $\bar{\lambda} \in \mathcal{D}$ and set the Littlewood decomposition $\Phi_n((\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}) = (\omega, \nu^{(0)}, \dots, \nu^{(n-1)})$. Then $\bar{\lambda}$ belongs to $C_n^{d,\mathrm{tr}}$ if and only if:

* $\nu^{(j)} = \emptyset$ for any $0 \le j \le n-2 \setminus \{\frac{n}{2}\}$,
* let $m = \max(\{1\} \cup \{(n + \bar{\lambda}_i)/n \mid \bar{\lambda}_i \equiv 0 \pmod n\})$. Then $\nu^{(n-1)}$ is the rectangle partition $(m-1) \times m$. Moreover, when $n$ is even, then $\nu^{(n/2-1)}$ is the $m' \times m'$ with

$$m' = \max(\{0\} \cup \{\bar{\lambda}_i/n \mid \bar{\lambda}_i \equiv n/2 \pmod n\})$$

or equivalently $m' = \max(k+1 \mid c_{kn+n/2-1} = 0)$.

In addition, by setting $\phi_{2n-2}(\omega) = (m_0, \dots, m_{n-3}, 0, -m_{n-3}, \dots, -m_0, 0)$, we get

$$(33) \qquad |\bar{\lambda}| = (2n-2) \left( \sum_{i=0}^{n-3} m_i^2 + m^2 + m'^2 \right) + \left( -(2n-2)m + \sum_{i=0}^{n-2} 2(i-n+1)m_i \right).$$

The foregoing subsections use the same arguments. We first use the result from Section 5 giving the correspondence between the affine Grassmannian elements and some subsets of self-conjugate $(2n)$-cores. Next, by considering the bijection described in Theorem 6.9, we exhibit the correspondence between the affine Grassmannian elements and some subsets of integer partitions so that the atomic length of any affine Grassmannian element coincides with the weight (i.e. the number of boxes) of the corresponding partition.

**6.4. Type $C_n^{(1)}$.** This case is easy since for any $c \in W_{C_n^{(1)}}^0$ identified to $C_{2n}^s$, we have $L_{C_n^{(1)}}(c) = L_{A_{2n-1}^{(1)}}(c) = |c|$ i.e. the atomic length is given by the number of boxes of $c$ regarded as a self-conjugate $2n$-core. Therefore as already seen in (29)

$$(34) \qquad L_{C_n^{(1)}}(c) = 2n \sum_{i=0}^{n-1} m_i^2 + \sum_{i=0}^{n-1} (2i - 2n + 1)m_i = 2n \sum_{i=0}^{n-1} \beta_i^2 - \sum_{i=0}^{n-1} (2i - 1)\beta_i.$$

Observe also that we have $a_0 = \sum_{i=0}^{n-1} \beta_i^2$ and $a_n = a_0 - \beta_1 - \dots - \beta_n$ from which it becomes easy to compute the atomic length in any type from the results of Section 5.

**6.5. Type $D_{n+1}^{(2)}$.** Recall from Proposition 5.5, that if $c \in W_{D_{n+1}^{(2)}}^0 = C_{2n}^s$, we get:

$$L_{D_{n+1}^{(2)}}(c) = \frac{1}{2} \left( (2n+2) \sum_{i=1}^{n} \beta_i^2 - 2i \beta_i \right) = (n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} i \beta_i \\ = (n+1) \sum_{i=0}^{n-1} m_i^2 + \sum_{i=0}^{n-1} (i-n)m_i.$$

Set $c \in W_{D_{n+1}^{(2)}}^0 = C_{2n}^s$ and $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ (see Definition 6.8). Set $c' = \phi_{2n+2}^{-1}(0, m_0, \dots, m_{n-1}, 0)$.

One easily sees by using that $L_{D_{n+1}^{(2)}}(c) = \frac{1}{2}L_{C_n^{(1)}}(c')$ which suggest that $c$ could be replaced by $c'$ in order to get an atomic length counted by the number of boxes in a partition. In fact, we have the following stronger statement:

**Proposition 6.17.** The core $c'$ belongs to $C_{2n+2}^{dd}$ and there exists $\bar{\lambda} \in C_{2n+2}^{d,r}$ such that $c' = \bar{\lambda}\bar{\lambda}$. Moreover, the map which associates to any $c \in C_{2n}^s$ the so obtained distinct partition $\bar{\lambda} \in C_{2n+2}^{d,r}$ is a bijection such that $L_{D_{n+1}^{(2)}}(c) = |\bar{\lambda}|$.

*Proof.* The first sentence of the proposition is a consequence of Corollary 6.12 (even case) and Lemma 6.13. Now the map which sends $c \in C_{2n}^s$ to $\bar{\lambda} \in C_{2n+2}^{d,r}$ is bijective by composition. The equality $L(c) = |\bar{\lambda}|$ derives from Proposition 6.14, (30) and (29) setting $\beta_i = m_{2n+2-i}$. $\square$

**Example 6.18.** Take $n=2$ and $c = (3, 2, 1) \in W_{D_3^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1, -1, 1, -1)$. Then we obtain $(c_6')^{-1}(0, 1, -1, 0, 1, -1)$ (5, 3, 1, 1) and $\bar{\lambda} = (4, 1)$. We have $L_{D_{n+1}^{(2)}}(c) = |c'| = 5$.

**6.6. Type $A_{2n}^{(2)}$.** Recall from Proposition 5.6, that if $c \in W_{A_{2n}^{(2)}}^0 = C_{2n}^s$, then by using (28), one derives:

$$L_{A_{2n}^{(2)}}(c) = \frac{1}{2} \left( (2n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (2i-1)\beta_i \right).$$

One can observe that in this case the coefficient of the monomial $\beta_n$ is $-(2n+1)/2$ which happens to be the opposite of the coefficient of $\beta_n^2$. Set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ and $c' = \phi_{2n+1}^{-1}(0, m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$
• let $m = \max(\{1\} \cup \{(n + \bar{\lambda}_i)/n \mid \bar{\lambda}_i \equiv 0 \pmod n\})$. Then $\nu^{(n-1)}$ is the rectangle partition $(m-1) \times m$. Moreover, when $n$ is even, then $\nu^{(n/2-1)}$ is the $m' \times m'$ with

$$m' = \max(\{0\} \cup \{\bar{\lambda}_i/n \mid \bar{\lambda}_i \equiv n/2 \pmod n\})$$

or equivalently $m' = \max(k+1 \mid c_{kn+n/2-1} = 0)$.

In addition, by setting $\phi_{2n-2}(\omega) = (m_0, \dots, m_{n-3}, 0, -m_{n-3}, \dots, -m_0, 0)$, we get

$$(33) \qquad |\bar{\lambda}| = (2n-2) \left( \sum_{i=0}^{n-3} m_i^2 + m^2 + m'^2 \right) + \left( -(2n-2)m + \sum_{i=0}^{n-2} 2(i-n+1)m_i \right).$$

The foregoing subsections use the same arguments. We first use the result from Section 5 giving the correspondence between the affine Grassmannian elements and some subsets of self-conjugate (2n)-cores. Next, by considering the bijection described in Theorem 6.9, we exhibit the correspondence between the affine Grassmannian elements and some subsets of integer partitions so that the atomic length of any affine Grassmannian element coincides with the weight (i.e. the number of boxes) of the corresponding partition.

**6.4. Type $C_n^{(1)}$.** This case is easy since for any $c \in W_{C_n^{(1)}}^0$ identified to $C_{2n}^s$, we have $L_{C_n^{(1)}}(c) = L_{A_{2n-1}^{(1)}}(c) = |c|$ i.e. the atomic length is given by the number of boxes of $c$ regarded as a self-conjugate 2n-core. Therefore as already seen in (29)

$$(34) \qquad L_{C_n^{(1)}}(c) = 2n \sum_{i=0}^{n-1} m_i^2 + \sum_{i=0}^{n-1} (2i - 2n + 1)m_i = 2n \sum_{i=0}^{n-1} \beta_i^2 - \sum_{i=0}^{n-1} (2i - 1)\beta_i.$$

Observe also that we have $a_0 = \sum_{i=0}^{n-1} \beta_i^2$ and $a_n = a_0 - \beta_1 - \dots - \beta_n$ from which it becomes easy to compute the atomic length in any type from the results of Section 5.

**6.5. Type $D_{n+1}^{(2)}$.** Recall from Proposition 5.5, that if $c \in W_{D_{n+1}^{(2)}}^0 = C_{2n}^s$, we get:

$$L_{D_{n+1}^{(2)}}(c) = \frac{1}{2} \left( (2n+2) \sum_{i=1}^{n} \beta_i^2 - 2i \beta_i \right) = (n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} i \beta_i \\ = (n+1) \sum_{i=0}^{n-1} m_i^2 + \sum_{i=0}^{n-1} (i-n)m_i.$$

Set $c \in W_{D_{n+1}^{(2)}}^0 = C_{2n}^s$ and $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ (see Definition 6.8). Set $c' = \phi_{2n+2}^{-1}(0, m_0, \dots, m_{n-1}, 0)$.

One easily sees by using that $L_{D_{n+1}^{(2)}}(c) = \frac{1}{2} L_{C_n^{(1)}}(c')$ which suggest that $c$ could be replaced by $c'$ in order to get an atomic length counted by the number of boxes in a partition. In fact, we have the following stronger statement:

**Proposition 6.17.** The core $c'$ belongs to $C_{2n+2}^{dd}$ and there exists $\bar{\lambda} \in C_{2n+2}^{d,r}$ such that $c' = \bar{\lambda}\bar{\lambda}$. Moreover, the map which associates to any $c \in C_{2n}^s$ the so obtained distinct partition $\bar{\lambda} \in C_{2n+2}^{d,r}$ is a bijection such that $L_{D_{n+1}^{(2)}}(c) = |\bar{\lambda}|$.

*Proof.* The first sentence of the proposition is a consequence of Corollary 6.12 (even case) and Lemma 6.13. Now the map which sends $c \in C_{2n}^s$ to $\bar{\lambda} \in C_{2n+2}^{d,r}$ is bijective by composition. The equality $L(c) = |\bar{\lambda}|$ derives from Proposition 6.14, (30) and (29) setting $\beta_i = m_{2n+2-i}$. $\square$

**Example 6.18.** Take $n=2$ and $c = (3, 2, 1) \in W_{D_3^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1, -1, 1, -1)$. Then we obtain $(c_6')^{-1}(0, 1, -1, 0, 1, -1)$ (5, 3, 1, 1) and $\bar{\lambda} = (4, 1)$. We have $L_{D_{n+1}^{(2)}}(c) = |c'| = 5$.

**6.6. Type $A_{2n}^{(2)}$.** Recall from Proposition 5.6, that if $c \in W_{A_{2n}^{(2)}}^0 = C_{2n}^s$, then by using (28), one derives:

$$L_{A_{2n}^{(2)}}(c) = \frac{1}{2} \left( (2n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (2i-1)\beta_i \right).$$

One can observe that in this case the coefficient of the monomial $\beta_n$ is $-(2n+1)/2$ which happens to be the opposite of the coefficient of $\beta_n^2$. Set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ and $c' = \phi_{2n+1}^{-1}(0, m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$
(see Definition 6.8). Using (30), $c'$ is a doubled distinct $(2n+1)$-core. Therefore, by Lemma 6.13, there exists a unique $\bar{\lambda} \in C_{2n+1}^{d,r}$ such that $c' = \bar{\lambda}\bar{\lambda}$. Moreover we have that:

$$L_{A_{2n}^{(2)}}(c) = |\bar{\lambda}|.$$

**Example 6.19.** Take $n=2$ and $c=(3,2,1) \in W_{A_4^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1,-1,1,-1)$. Then $c_5'^{-1}(0,1,-1,1,-1) = (4,3,1)$ and $\bar{\lambda} = (3,1)$. We have $L(c') = 8$.

**6.7. Type $A_{2n}^{(2)}$.** Recall from Proposition 5.7, that if $c \in W_{A_{2n}^{(2)}}^0 = C_{2n}^s$, then by using (28), one derives:

$$L_{A_{2n}^{(2)}}(c) = (2n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} 2i\beta_i.$$

Set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ and $c' = \phi_{2n+1}^{-1}(m_0, \dots, m_{n-1}, 0, -m_{n-1}, \dots, -m_0)$. The map sending $c \in C_n^s$ to $c' \in C_{2n+1}^s$ is a bijection. Moreover, by Corollary 6.11, we have:

$$L_{A_{2n}^{(2)}}(c) = |c'|.$$

**Example 6.20.** Take $n=2$ and $c = (3,2,1) \in W_{A_4^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1,-1,1,-1)$. Then $c' = \phi_5^{-1}(1,-1,0,1,-1) = (4,2,1,1)$. We have $L(c) = |c'| = 8$.

**6.8. Type $B_n^{(1)}$.** For all the remaining types in this section, the affine Grassmannian elements are in bijection with the subset $C_{2n}^{s,p}$ of self-conjugate $(2n)$-cores with an even diagonal. So take $c \in C_{2n}^{s,p}$ and set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$. A core has an even diagonal if and only if the number of horizontal steps after the corner of the Durfee square is even. This is equivalent to the fact that the number of letters "0" of positive index in the corresponding word $\psi(c)$ is even. Therefore $c \in C_{2n}^{s,p}$ is equivalent to

$$\sum_{i=0}^{n-1} |m_i| = \sum_{i=0}^{n-1} |\beta_i| \equiv 0 \pmod{2}.$$

One derives from Proposition 5.9 that for any $c \in W_{B_n^{(1)}}^0 = C_{2n}^{s,p}$

$$L_{B_n^{(1)}}(c) = \frac{1}{2} \left( 2n \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} 2i\beta_i \right) = n \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} i\beta_i.$$

Consider $\bar{\lambda} \in C_{2n}^{d,\mathrm{tr},r}$ and set $\Phi_{2n}((\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}) = (\omega, \nu^{(0)}, \dots, \nu^{(n-1)})$. By Proposition 6.15, we obtain $\phi_{2n}(\omega) = (m_0, \dots, m_{n-2}, 0, -m_0, \dots, -m_{n-2})$ and $\nu^{(n-1)}$ is a rectangle $(m-1) \times m$ with $m \in \mathbb{N}^*$. This permits to define a bijection $f_{2n}$ associating to each $\bar{\lambda}$ its corresponding vector $(m, m_0, \dots, m_{n-2}) \in \mathbb{N}^* \times \mathbb{Z}^{n-1}$. We can now consider the map

$$g: \begin{array}{rcl} \mathbb{N}^* \times \mathbb{Z}^{n-1} & \to & \mathbb{Z}^{2n} \\ (m, m_0, \dots, m_{n-2}) & \mapsto & \begin{cases} (m, m_0, \dots, m_{n-2}, -m_{n-2}, \dots, -m_0, -m) & \text{if } m \equiv \sum_{i=0}^{n-2} m_i \pmod{2}, \\ (-m+1, m_0, \dots, m_{n-2}, -m_{n-2}, \dots, -m_0, m-1) & \text{otherwise.} \end{cases} \end{array}$$

which is a bijection from $\mathbb{N}^* \times \mathbb{Z}^{n-1}$ to

$$\{(m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0) \in \mathbb{Z}^{2n} \mid \sum_{i=0}^{n-1} |m_i| \equiv 0 \pmod{2}\}.$$

So, the map $\phi_{2n}^{-1} \circ g \circ f_{2n}$ is the desired bijection from $C_{2n}^{d,\mathrm{tr},r}$ to $C_{2n}^{s,p}$. Moreover by Proposition 6.15, by setting $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n})^{-1}(c)$, we get:

$$L_{B_n^{(1)}}(c) = |\bar{\lambda}|.$$

**Example 6.21.** Take $n=3$ and $c = (10, 6, 6, 4, 3, 3, 1, 1, 1, 1) \in W_{B_3^{(1)}}^0 = C_6^{s,p}$ and $\phi_6(c) = (1, -1, -2, 2, 1, -1)$. Then $g^{-1} \circ \phi_6(c) = (1, -1, -2)$. Then $c' = (10, 6, 6, 3, 3, 3, 3, 1, 1, 1)$ and $\bar{\lambda} = (10, 5, 4)$. We have $L_{B_3^{(1)}}(c) = |\bar{\lambda}| = 19$.
(see Definition 6.8). Using (30), $c'$ is a doubled distinct $(2n+1)$-core. Therefore, by Lemma 6.13, there exists a unique $\bar{\lambda} \in C_{2n+1}^{d,r}$ such that $c' = \bar{\lambda}\bar{\lambda}$. Moreover we have that:

$$L_{A_{2n}^{(2)}}(c) = |\bar{\lambda}|.$$

**Example 6.19.** Take $n=2$ and $c = (3,2,1) \in W_{A_4^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1,-1,1,-1)$. Then $c_5'^{-1}(0,1,-1,1,-1) = (4,3,1)$ and $\bar{\lambda} = (3,1)$. We have $L(c') = 8$.

**6.7. Type $A_{2n}^{(2)}$.** Recall from Proposition 5.7, that if $c \in W_{A_{2n}^{(2)}}^0 = C_{2n}^s$, then by using (28), one derives:

$$L_{A_{2n}^{(2)}}(c) = (2n+1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} 2i\beta_i.$$

Set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$ and $c' = \phi_{2n+1}^{-1}(m_0, \dots, m_{n-1}, 0, -m_{n-1}, \dots, -m_0)$. The map sending $c \in C_n^s$ to $c' \in C_{2n+1}^s$ is a bijection. Moreover, by Corollary 6.11, we have:

$$L_{A_{2n}^{(2)}}(c) = |c'|.$$

**Example 6.20.** Take $n=2$ and $c = (3,2,1) \in W_{A_4^{(2)}}^0 = C_4^s$ and $\phi_4(c) = (1,-1,1,-1)$. Then $c' = \phi_5^{-1}(1,-1,0,1,-1) = (4,2,1,1)$. We have $L(c) = |c'| = 8$.

**6.8. Type $B_n^{(1)}$.** For all the remaining types in this section, the affine Grassmannian elements are in bijection with the subset $C_{2n}^{s,p}$ of self-conjugate $(2n)$-cores with an even diagonal. So take $c \in C_{2n}^{s,p}$ and set $\phi_{2n}(c) = (m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0)$. A core has an even diagonal if and only if the number of horizontal steps after the corner of the Durfee square is even. This is equivalent to the fact that the number of letters "0" of positive index in the corresponding word $\psi(c)$ is even. Therefore $c \in C_{2n}^{s,p}$ is equivalent to

$$\sum_{i=0}^{n-1} |m_i| = \sum_{i=0}^{n-1} |\beta_i| \equiv 0 \pmod{2}.$$

One derives from Proposition 5.9 that for any $c \in W_{B_n^{(1)}}^0 = C_{2n}^{s,p}$

$$L_{B_n^{(1)}}(c) = \frac{1}{2} \left( 2n \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} 2i\beta_i \right) = n \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} i\beta_i.$$

Consider $\bar{\lambda} \in C_{2n}^{d,\mathrm{tr},r}$ and set $\Phi_{2n}((\bar{\lambda}\bar{\lambda})^{\mathrm{tr}}) = (\omega, \nu^{(0)}, \dots, \nu^{(n-1)})$. By Proposition 6.15, we obtain $\phi_{2n}(\omega) = (m_0, \dots, m_{n-2}, 0, -m_0, \dots, -m_{n-2})$ and $\nu^{(n-1)}$ is a rectangle $(m-1) \times m$ with $m \in \mathbb{N}^*$. This permits to define a bijection $f_{2n}$ associating to each $\bar{\lambda}$ its corresponding vector $(m, m_0, \dots, m_{n-2}) \in \mathbb{N}^* \times \mathbb{Z}^{n-1}$. We can now consider the map

$$g: \begin{array}{ccc} \mathbb{N}^* \times \mathbb{Z}^{n-1} & \to & \mathbb{Z}^{2n} \\ (m, m_0, \dots, m_{n-2}) & \mapsto & \begin{cases} (m, m_0, \dots, m_{n-2}, -m_{n-2}, \dots, -m_0, -m) & \text{if } m \equiv \sum_{i=0}^{n-2} m_i \pmod{2}, \\ (-m+1, m_0, \dots, m_{n-2}, -m_{n-2}, \dots, -m_0, m-1) & \text{otherwise.} \end{cases} \end{array}$$

which is a bijection from $\mathbb{N}^* \times \mathbb{Z}^{n-1}$ to

$$\{(m_0, \dots, m_{n-1}, -m_{n-1}, \dots, -m_0) \in \mathbb{Z}^{2n} \mid \sum_{i=0}^{n-1} |m_i| \equiv 0 \pmod{2}\}.$$

So, the map $\phi_{2n}^{-1} \circ g \circ f_{2n}$ is the desired bijection from $C_{2n}^{d,\mathrm{tr},r}$ to $C_{2n}^{s,p}$. Moreover by Proposition 6.15, by setting $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n})^{-1}(c)$, we get:

$$L_{B_n^{(1)}}(c) = |\bar{\lambda}|.$$

**Example 6.21.** Take $n=3$ and $c = (10, 6, 6, 4, 3, 3, 1, 1, 1, 1) \in W_{B_3^{(1)}}^0 = C_6^{s,p}$ and $\phi_6(c) = (1, -1, -2, 2, 1, -1)$. Then $g^{-1} \circ \phi_6(c) = (1, -1, -2)$. Then $c' = (10, 6, 6, 3, 3, 3, 3, 1, 1, 1)$ and $\bar{\lambda} = (10, 5, 4)$. We have $L_{B_3^{(1)}}(c) = |\bar{\lambda}| = 19$.
6.9. **Type $A_{2n-1}^{(2)}$.** Recall from Proposition 5.11 that if $c \in W_{A_{2n-1}^{(2)}}^{0} = C_{2n}^{s,p}$, we have:

$$L_{A_{2n-1}^{(2)}}(c) = \frac{1}{2} \left( (2n-1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (2i-1) \beta_i \right).$$

As in the previous subsection, by Proposition 6.15, $\bar{\lambda} \in C_{2n-1}^{d,\text{tr}}$ is mapped bijectively to the vector $(m, m_0, \dots, m_{n-2}) \in \mathbb{N}^* \times \mathbb{Z}^{n-1}$. Let us denote this bijection by $f_{2n-1}$.

Therefore $c$ is mapped bijectively to $\bar{\lambda} \in C_{2n-1}^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n-1})^{-1}(c)$. Moreover

$$L_{A_{2n-1}^{(2)}}(c) = |\bar{\lambda}|.$$

**Example 6.22.** Take $n=3$ and $c = (10, 6, 4, 3, 3, 1, 1, 1, 1) \in W_{A_5}^{0} = C_6^{s,p}$ and $\phi_6(c) = (1, -1, -2, 2, 1, -1)$. Then $g^{-1} \circ \phi_6(c) = (1, -1, -2)$. Then $c' = (8, 5, 5, 3, 3, 3, 1, 1, 1)$ and $\bar{\lambda} = (8, 4, 3)$. We have $L_{A_5^{(2)}}(c) = |\bar{\lambda}| = 15$.

6.10. **Type $D_n^{(1)}$.** Recall from Proposition 5.12 that if $c \in W_{D_n^{(1)}}^{0} = C_{2n}^{s,p}$, we have then by using (28)

$$L_{D_n^{(1)}}(c) = (n-1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (i-1)\beta_i.$$

By Proposition 6.16, $\bar{\lambda} \in C_{2n}^{d,\text{tr}}$ is mapped bijectively to the vector $(m, m_0, \dots, m_{n-3}, m') \in \mathbb{N}^* \times \mathbb{Z}^{n-1} \times \mathbb{N}$. Let us denote this bijection by $f_{2n-2}$. Therefore $c$ is mapped bijectively to $\bar{\lambda} \in C_{2n-2}^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n-2})^{-1}(c)$. Moreover

$$L_{D_n^{(1)}}(c) = |\bar{\lambda}|.$$

6.11. **Type $D_4^{(3)}$.** From Proposition 5.15, for any $c \in W_{D_4}^{0}$, note that $L_{D_4^{(3)}}(c)$ has the same expression as the atomic length of an element of type $D_3^{(1)}$, which does not exist. Nevertheless the subset $C_{2n-2}^{d,\text{tr}}$ is well-defined for $n=3$: it is the subset of distinct partitions such that $4 \notin \mathcal{H}(\bar{\lambda}^{\text{tr}}) = \{h_s, s \in \bar{\lambda}\bar{\lambda}, \varepsilon = -\}$. Moreover the equation (18) implies that $\beta_3 = \beta_1 - \beta_2$. By Proposition 6.16, $c \in W_{D_4}^{0} = C_6^g$ is mapped bijectively to $\bar{\lambda} \in C_4^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_6^{-1} \circ g \circ f_4)^{-1}(c)$ such that $m = \pm m' \pm m_0 - \varphi$, where $m, m'$ and $m_0$ are as defined in Proposition 6.16 and $\varphi \in \{0, 1\}$. Moreover

$$L_{D_4^{(3)}}(c) = |\bar{\lambda}|.$$

6.12. **Type $G_2^{(1)}$.** Similarly to the previous subsection, from Proposition 5.16), note that this time for any $c \in W_{G_2^{(1)}}^{0}$, $L_{G_2^{(1)}}(c)$ has the same expression as the atomic length of an element type $B_3^{(1)}$ with an additional restriction.

By Proposition 6.15, $c \in W_{G_2^{(1)}}^{0}$ is mapped bijectively to $\bar{\lambda} = (\phi_6^{-1} \circ g \circ f_6)^{-1}(c)$ with the additional condition that $m$ is equal to $m_0 + m_1$ if $m \equiv m_0 + m_1 \pmod 2$ or to $-m_0 - m_1 - 1$ otherwise, where $m, m_0, m_1$ are defined as in Proposition 6.15. Moreover

$$L_{G_2^{(1)}}(c) = |\bar{\lambda}|.$$

## 7. MORE ON n-CORES AND THEIR n-ABACUS

The goal of this section is first to use the connection between the affine Grassmannian elements and the combinatorial models described in Section 6 to compute $\varepsilon(c)$ as its appears in Theorem 4.3. We next reinterpret the dominant weights appearing in the Weyl characters of the decomposition obtained in Theorem 4.3. These results can be deduced from those obtained in [Wah23] and we omit the proofs for short.

Take $n$ a positive integer. Let us introduce

$$
\begin{align*}
H_n(\lambda) &:= \{s \in \lambda \mid h_s < n\}, \\
H_n(\bar{\lambda}) &:= \{h_s \in \mathcal{H}(\bar{\lambda}) \mid h_s < n\}, \\
H_n(\bar{\lambda}^{\text{tr}}) &:= \{h_s \in \mathcal{H}(\bar{\lambda}^{\text{tr}}) \mid h_s < n\}, \\
H_n(\bar{\lambda} \cup \Delta) &:= \{h_s \in \mathcal{H}(\bar{\lambda}) \uplus \{2\bar{\lambda}_1, \dots, 2\bar{\lambda}_\ell\} \mid h_s < n\}, \\
H_n(\bar{\lambda}^{\text{tr}} \cup \Delta) &:= \{h_s \in \mathcal{H}(\bar{\lambda}^{\text{tr}}) \uplus \{2\bar{\lambda}_1, \dots, 2\bar{\lambda}_\ell\} \mid h_s < n\}.
\end{align*}
$$
6.9. **Type $A_{2n-1}^{(2)}$.** Recall from Proposition 5.11 that if $c \in W_{A_{2n-1}^{(2)}}^{0} = C_{2n}^{s,p}$, we have:

$$L_{A_{2n-1}^{(2)}}(c) = \frac{1}{2} \left( (2n-1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (2i-1) \beta_i \right).$$

As in the previous subsection, by Proposition 6.15, $\bar{\lambda} \in C_{2n-1}^{d,\text{tr}}$ is mapped bijectively to the vector $(m, m_0, \dots, m_{n-2}) \in \mathbb{N}^* \times \mathbb{Z}^{n-1}$. Let us denote this bijection by $f_{2n-1}$.

Therefore $c$ is mapped bijectively to $\bar{\lambda} \in C_{2n-1}^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n-1})^{-1}(c)$. Moreover

$$L_{A_{2n-1}^{(2)}}(c) = |\bar{\lambda}|.$$

**Example 6.22.** Take $n=3$ and $c = (10, 6, 6, 4, 3, 3, 1, 1, 1, 1) \in W_{A_5}^{0} = C_6^{s,p}$ and $\phi_6(c) = (1, -1, -2, 2, 1, -1)$. Then $g^{-1} \circ \phi_6(c) = (1, -1, -2)$. Then $c' = (8, 5, 5, 3, 3, 3, 1, 1, 1)$ and $\bar{\lambda} = (8, 4, 3)$. We have $L_{A_5^{(2)}}(c) = |\bar{\lambda}| = 15$.

6.10. **Type $D_n^{(1)}$.** Recall from Proposition 5.12 that if $c \in W_{D_n^{(1)}}^{0} = C_{2n}^{s,p}$, we have then by using (28)

$$L_{D_n^{(1)}}(c) = (n-1) \sum_{i=1}^{n} \beta_i^2 - \sum_{i=1}^{n} (i-1)\beta_i.$$

By Proposition 6.16, $\bar{\lambda} \in C_{2n}^{d,\text{tr}}$ is mapped bijectively to the vector $(m, m_0, \dots, m_{n-3}, m') \in \mathbb{N}^* \times \mathbb{Z}^{n-1} \times \mathbb{N}$. Let us denote this bijection by $f_{2n-2}$. Therefore $c$ is mapped bijectively to $\bar{\lambda} \in C_{2n-2}^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_{2n}^{-1} \circ g \circ f_{2n-2})^{-1}(c)$. Moreover

$$L_{D_n^{(1)}}(c) = |\bar{\lambda}|.$$

6.11. **Type $D_4^{(3)}$.** From Proposition 5.15, for any $c \in W_{D_4^{(3)}}^{0}$, note that $L_{D_4^{(3)}}(c)$ has the same expression as the atomic length of an element of type $D_3^{(1)}$, which does not exist. Nevertheless the subset $C_{2n-2}^{d,\text{tr}}$ is well-defined for $n=3$: it is the subset of distinct partitions such that $4 \notin \mathcal{H}(\bar{\lambda}^{\text{tr}}) = \{h_s, s \in \bar{\lambda}\bar{\lambda}, \varepsilon = -\}$. Moreover the equation (18) implies that $\beta_3 = \beta_1 - \beta_2$. By Proposition 6.16, $c \in W_{D_4^{(3)}}^{0} = C_6^g$ is mapped bijectively to $\bar{\lambda} \in C_4^{d,\text{tr}}$ with $\bar{\lambda} = (\phi_6^{-1} \circ g \circ f_4)^{-1}(c)$ such that $m = \pm m' \pm m_0 - \varphi$, where $m, m'$ and $m_0$ are as defined in Proposition 6.16 and $\varphi \in \{0, 1\}$. Moreover

$$L_{D_4^{(3)}}(c) = |\bar{\lambda}|.$$

6.12. **Type $G_2^{(1)}$.** Similarly to the previous subsection, from Proposition 5.16), note that this time for any $c \in W_{G_2^{(1)}}^{0}$, $L_{G_2^{(1)}}(c)$ has the same expression as the atomic length of an element type $B_3^{(1)}$ with an additional restriction.

By Proposition 6.15, $c \in W_{G_2^{(1)}}^{0}$ is mapped bijectively to $\bar{\lambda} = (\phi_6^{-1} \circ g \circ f_6)^{-1}(c)$ with the additional condition that $m$ is equal to $m_0 + m_1$ if $m \equiv m_0 + m_1 \pmod 2$ or to $-m_0 - m_1 - 1$ otherwise, where $m, m_0, m_1$ are defined as in Proposition 6.15. Moreover

$$L_{G_2^{(1)}}(c) = |\bar{\lambda}|.$$

## 7. MORE ON $n$-CORES AND THEIR $n$-ABACUS

The goal of this section is first to use the connection between the affine Grassmannian elements and the combinatorial models described in Section 6 to compute $\varepsilon(c)$ as its appears in Theorem 4.3. We next reinterpret the dominant weights appearing in the Weyl characters of the decomposition obtained in Theorem 4.3. These results can be deduced from those obtained in [Wah23] and we omit the proofs for short.

Take $n$ a positive integer. Let us introduce

$$
\begin{align*}
H_n(\lambda) &:= \{s \in \lambda \mid h_s < n\}, \\
H_n(\bar{\lambda}) &:= \{h_s \in \mathcal{H}(\bar{\lambda}) \mid h_s < n\}, \\
H_n(\bar{\lambda}^{\text{tr}}) &:= \{h_s \in \mathcal{H}(\bar{\lambda}^{\text{tr}}) \mid h_s < n\}, \\
H_n(\bar{\lambda} \cup \Delta) &:= \{h_s \in \mathcal{H}(\bar{\lambda}) \uplus \{2\bar{\lambda}_1, \dots, 2\bar{\lambda}_\ell\} \mid h_s < n\}, \\
H_n(\bar{\lambda}^{\text{tr}} \cup \Delta) &:= \{h_s \in \mathcal{H}(\bar{\lambda}^{\text{tr}}) \uplus \{2\bar{\lambda}_1, \dots, 2\bar{\lambda}_\ell\} \mid h_s < n\}.
\end{align*}
$$
We can in fact precise the results of the previous section and use the sets above to compute the signature of the affine Grassmannian elements as detailed in Table 1.

<table><caption>TABLE 1. Table of affine types with their corresponding partition family</caption><thead><tr><th>T</th><th>C<sub>T</sub></th><th>&epsilon;(c)</th></tr></thead><tbody><tr><td>A<sub>n-1</sub><sup>(1)</sup></td><td>C<sub>n</sub></td><td>(-1)<sup>#H<sub>n</sub>(&lambda;)</sup></td></tr><tr><td>B<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n</sub><sup>d,tr,r</sup></td><td>(-1)<sup>#H<sub>2n</sub>(&lambda;<sup>tr</sup>&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>A<sub>2n-1</sub><sup>(2)</sup></td><td>C<sub>2n-1</sub><sup>d,tr,r</sup></td><td>(-1)<sup>#H<sub>2n-1</sub>(&lambda;)<sup>tr</sup></sup></td></tr><tr><td>C<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n</sub><sup>s</sup></td><td>(-1)<sup>#H<sub>2n</sub>(&lambda;)</sup></td></tr><tr><td>D<sub>n+1</sub><sup>(2)</sup></td><td>C<sub>2n+2</sub><sup>d,r</sup></td><td>(-1)<sup>#H<sub>2n+2</sub>(&lambda;&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>A<sub>2n</sub><sup>(2)</sup></td><td>C<sub>2n+1</sub><sup>s</sup></td><td>(-1)<sup>#H<sub>2n+1</sub>(&lambda;)</sup></td></tr><tr><td>A<sub>2n</sub><sup>'(2)</sup></td><td>C<sub>2n+1</sub><sup>d,r</sup></td><td>(-1)<sup>#H<sub>2n+1</sub>(&lambda;&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>D<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n-2</sub><sup>d,tr</sup></td><td>(-1)<sup>#H<sub>2n-2</sub>(&lambda;<sup>tr</sup>&cup;&Delta;)</sup></td></tr></tbody></table>

We recall here the following definition that bridges the abacus model appearing in [Wah23] and the affine Grassmannian elements.

**Definition 7.1.** Let $n$ and $g$ be two positive integers such that $n \le g$. Set $\lambda \in \mathcal{P}$ and $\psi(\lambda) = (c_k)_{k \in \mathbb{Z}}$ its corresponding binary word. For $i \in \{0, \dots, g-1\}$, define $m_i := \max\{(k+1)g + i \mid c_{kg+i} = 0\}$. Let $\sigma: \{1, \dots, g\} \to \{0, \dots, g-1\}$ be the unique bijection such that $\beta_{\sigma(1)} > \dots > \beta_{\sigma(g)}$. The vector $\mathbf{v} := (\beta_{\sigma(1)}, \dots, \beta_{\sigma(n)})$ is called the $V_{g,n}$-coding corresponding to $\lambda$.

By definition of the elements of $C_T$ in Table 1 and Proposition 6.14, one has the following lemma.

**Lemma 7.2.** Take $T$ in Table 1 such that $T \ne A_{n-1}^{(1)}$ and $\bar{\lambda} \in C_T = C_g^a$. Let $\mu$ be the corresponding partition in either $DD$ (if a is $d, r$), $DD^{\mathrm{tr}}$ (if a is $d, \mathrm{tr}, r$ or a is $d, \mathrm{tr}$) or $SC$ (in the other cases). Let $\mathbf{v} := (\beta_{\sigma(1)}, \dots, \beta_{\sigma(n)})$ be the $V_{g,n}$-coding corresponding to $\mu$. Then for any $1 \le i \le n$ such that $v_i > g$, we have:

* if $\mu$ is a doubled distinct partition, $v_i = \max\{\bar{\lambda}_k + g, k \ge 1, \bar{\lambda}_k \equiv \sigma(i) (\mathrm{mod} g)\}$

* otherwise, $v_i = \max\{\bar{\lambda}_k - 1, k \ge 1, \bar{\lambda}_k - 1 + g \equiv \sigma(i) (\mathrm{mod} g)\}$.

Recall from Theorem 4.3 that if $c \in W_a^0$, we can set $c = ut_\nu$ with $\nu \in M^* \cap (-P_+)$ and $u \in W^\nu$ for any element of the affine Grassmannian $W_a^0$. Moreover, each weight $-\eta^\vee\nu + u^{-1}(\hat{\rho}) - \hat{\rho}$ belongs to $P_+$, the set of dominants weights for $\tilde{A}$. The purpose of the following theorem is to bridge the gaps between our reformulation of the Weyl-Kac formula as a sum over elements of the affine Grassmannian and the formulation of the same formula within the language of multivariate series and $\theta$ function (see [RS06, Proposition 6.1]), where the sum is over $\mathbb{Z}^n$. Given Remark 4.2, we set for any affine algebra $\mathfrak{g}$ of rank $n$ from Table 2:

$$ \tilde{\eta}^{\vee} = \begin{cases} \eta^{\vee} & \text{if } \mathfrak{g} \neq C_n^{(1)}, \\ 2\eta^{\vee} & \text{if } \mathfrak{g} = C_n^{(1)}. \end{cases} $$

Note that $\tilde{\eta}^{\vee}$ corresponds to the parameter $g$ in [Mac72, Wah23].

**Theorem 7.3.** With the previous notation the one-to-one correspondences described in Section 6 associate to any affine Grassmannian element $c$ its corresponding partition $\lambda$ of $C_T$ in Table 1 so that by writing $\mathbf{v} = (v_i)_{1 \le i \le n}$ for the $V_{\tilde{\eta}^{\vee},n}$ coding of $\lambda$, we get:

$$ -\tilde{\eta}^{\vee}\nu + u^{-1}(\hat{\rho}) - \hat{\rho} = (v_i + i - \tilde{\eta}^{\vee})_{1 \le i \le n}. $$

Recall that the proof of the Nekrasov–Okounkov formula as performed by Han [Han08] heavily relies on a specialization of the Weyl-Kac in type $\tilde{A}_{n-1}^{(1)}$ and polynomiality argument. This approach can be extended to derive (u-analogues of) Nekrasov–Okounkov formulas for any classical affine type $T$ with an unbounded rank. In order to do so from Theorem 4.3, one needs to be able to convert the characters on the right-hand side of Theorem 4.3 (or more precisely their associated u-dimension formulas) as a product over hook lengths of elements of $C_T$. Actually one can reformulate the results from [Wah23, Section 4.2] for a set of Laurent variables $(X_k)_{k \in \mathbb{Z}}$ as the bridge between hook length products on the one hand and products over weights of the form $-\eta^\vee\nu + u^{-1}(\hat{\rho}) - \hat{\rho}$ associated with an affine Grassmannian element as in Theorem 4.3.
We can in fact precise the results of the previous section and use the sets above to compute the signature of the affine Grassmannian elements as detailed in Table 1.

<table><caption>TABLE 1. Table of affine types with their corresponding partition family</caption><thead><tr><th>T</th><th>C<sub>T</sub></th><th>&epsilon;(c)</th></tr></thead><tbody><tr><td>A<sub>n-1</sub><sup>(1)</sup></td><td>C<sub>n</sub></td><td>(-1)<sup>#H<sub>n</sub>(&lambda;)</sup></td></tr><tr><td>B<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n</sub><sup>d,tr,r</sup></td><td>(-1)<sup>#H<sub>2n</sub>(&lambda;<sup>tr</sup>&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>A<sub>2n-1</sub><sup>(2)</sup></td><td>C<sub>2n-1</sub><sup>d,tr,r</sup></td><td>(-1)<sup>#H<sub>2n-1</sub>(&lambda;)<sup>tr</sup></sup></td></tr><tr><td>C<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n</sub><sup>s</sup></td><td>(-1)<sup>#H<sub>2n</sub>(&lambda;)</sup></td></tr><tr><td>D<sub>n+1</sub><sup>(2)</sup></td><td>C<sub>2n+2</sub><sup>d,r</sup></td><td>(-1)<sup>#H<sub>2n+2</sub>(&lambda;&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>A<sub>2n</sub><sup>(2)</sup></td><td>C<sub>2n+1</sub><sup>s</sup></td><td>(-1)<sup>#H<sub>2n+1</sub>(&lambda;)</sup></td></tr><tr><td>A<sub>2n</sub><sup>'(2)</sup></td><td>C<sub>2n+1</sub><sup>d,r</sup></td><td>(-1)<sup>#H<sub>2n+1</sub>(&lambda;&cup;&Delta;)+&ell;(&lambda;)</sup></td></tr><tr><td>D<sub>n</sub><sup>(1)</sup></td><td>C<sub>2n-2</sub><sup>d,tr</sup></td><td>(-1)<sup>#H<sub>2n-2</sub>(&lambda;<sup>tr</sup>&cup;&Delta;)</sup></td></tr></tbody></table>

We recall here the following definition that bridges the abacus model appearing in [Wah23] and the affine Grassmannian elements.

**Definition 7.1.** Let $n$ and $g$ be two positive integers such that $n \le g$. Set $\lambda \in \mathcal{P}$ and $\psi(\lambda) = (c_k)_{k \in \mathbb{Z}}$ its corresponding binary word. For $i \in \{0, \dots, g-1\}$, define $m_i := \max\{(k+1)g + i \mid c_{kg+i} = 0\}$. Let $\sigma: \{1, \dots, g\} \to \{0, \dots, g-1\}$ be the unique bijection such that $\beta_{\sigma(1)} > \dots > \beta_{\sigma(g)}$. The vector $v := (\beta_{\sigma(1)}, \dots, \beta_{\sigma(n)})$ is called the $V_{g,n}$-coding corresponding to $\lambda$.

By definition of the elements of $C_T$ in Table 1 and Proposition 6.14, one has the following lemma.

**Lemma 7.2.** Take $T$ in Table 1 such that $T \ne A_{n-1}^{(1)}$ and $\bar{\lambda} \in C_T = C_g^a$. Let $\mu$ be the corresponding partition in either $DD$ (if a is $d, r$), $DD^{\text{tr}}$ (if a is $d, \text{tr}, r$ or a is $d, \text{tr}$) or $SC$ (in the other cases). Let $v := (\beta_{\sigma(1)}, \dots, \beta_{\sigma(n)})$ be the $V_{g,n}$-coding corresponding to $\mu$. Then for any $1 \le i \le n$ such that $v_i > g$, we have:

* if $\mu$ is a doubled distinct partition, $v_i = \max\{\bar{\lambda}_k + g, k \ge 1, \bar{\lambda}_k \equiv \sigma(i) (\text{mod } g)\}$

* otherwise, $v_i = \max\{\bar{\lambda}_k - 1, k \ge 1, \bar{\lambda}_k - 1 + g \equiv \sigma(i) (\text{mod } g)\}$.

Recall from Theorem 4.3 that if $c \in W_a^0$, we can set $c = ut_\nu$ with $\nu \in M^* \cap (-P_+)$ and $u \in W^\nu$ for any element of the affine Grassmannian $W_a^0$. Moreover, each weight $-\eta^\vee\nu + u^{-1}(\hat{\rho}) - \hat{\rho}$ belongs to $P_+$, the set of dominants weights for $\tilde{A}$. The purpose of the following theorem is to bridge the gaps between our reformulation of the Weyl-Kac formula as a sum over elements of the affine Grassmannian and the formulation of the same formula within the language of multivariate series and $\theta$ function (see [RS06, Proposition 6.1]), where the sum is over $\mathbb{Z}^n$. Given Remark 4.2, we set for any affine algebra $\mathfrak{g}$ of rank $n$ from Table 2:

$$ \tilde{\eta}^{\vee} = \begin{cases} \eta^{\vee} & \text{if } \mathfrak{g} \neq C_n^{(1)}, \\ 2\eta^{\vee} & \text{if } \mathfrak{g} = C_n^{(1)}. \end{cases} $$

Note that $\tilde{\eta}^{\vee}$ corresponds to the parameter $g$ in [Mac72, Wah23].

**Theorem 7.3.** With the previous notation the one-to-one correspondences described in Section 6 associate to any affine Grassmannian element $c$ its corresponding partition $\lambda$ of $C_T$ in Table 1 so that by writing $v = (v_i)_{1 \le i \le n}$ for the $V_{\tilde{\eta}^{\vee},n}$ coding of $\lambda$, we get:

$$ -\tilde{\eta}^{\vee}\nu + u^{-1}(\hat{\rho}) - \hat{\rho} = (v_i + i - \tilde{\eta}^{\vee})_{1 \le i \le n}. $$

Recall that the proof of the Nekrasov–Okounkov formula as performed by Han [Han08] heavily relies on a specialization of the Weyl-Kac in type $\tilde{A}_{n-1}^{(1)}$ and polynomiality argument. This approach can be extended to derive (u-analogues of) Nekrasov–Okounkov formulas for any classical affine type $T$ with an unbounded rank. In order to do so from Theorem 4.3, one needs to be able to convert the characters on the right-hand side of Theorem 4.3 (or more precisely their associated u-dimension formulas) as a product over hook lengths of elements of $C_T$. Actually one can reformulate the results from [Wah23, Section 4.2] for a set of Laurent variables $(X_k)_{k \in \mathbb{Z}}$ as the bridge between hook length products on the one hand and products over weights of the form $-\eta^\vee\nu + u^{-1}(\hat{\rho}) - \hat{\rho}$ associated with an affine Grassmannian element as in Theorem 4.3.
**Theorem 7.4.** Let $(X_k)_{k \in \mathbb{Z}}$ be formal variables. For any affine type $T$ of rank $n$ in Table 1 and any element of the affine Grassmannian $c \in W_a^0$ such that $c = ut_\nu$ with $\nu \in Q \cap (-P_+) \text{ and } u \in W^\nu$. Set $\mathbf{r} = (r_i)_{1 \le i \le n}$ the vector such that $r_i = (-\tilde{\eta}^\vee \nu + u^{-1}(\mathring{\rho}) - \mathring{\rho})_i$ for all $i$. Let $\lambda \in C_T$ and $\mathcal{H}(\lambda)_T$ its corresponding elements in Table 1 and Table 2, respectively. Set $\alpha_T(i) = \#\{h_s \in \mathcal{H}(\lambda)_T, h_s = \tilde{\eta}^\vee - i\}$ and where $\alpha'_T(i) = \alpha_T(i) + \#\{h \in \bar{\lambda}, 2h = \tilde{\eta}^\vee - i\} + \#\{h \in \bar{\lambda}, h = \tilde{\eta}^\vee - i\}$. Then we have:

* if $T = A_{n-1}^{(1)}$

$$ (35) \qquad \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in C_n^{(1)} \cup D_{n+1}^{(2)} \cup A_{2n}^{(2)} \cup A'_{2n}^{(2)}$:

$$ (36) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h-\tilde{\eta}^{\vee}} X_{h-\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in B_n^{(1)} \cup A_{2n-1}^{(2)}$:

$$ (37) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in D_n^{(1)}$:

$$ (38) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h+\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

<table><caption>TABLE 2. Table of affine types with their corresponding hook length product</caption><thead><tr><th>T</th><th>H(λ)<sub>T</sub></th><th>Δ<sub>T</sub>(r)</th></tr></thead><tbody><tr><td>A<sub>n-1</sub><sup>(1)</sup></td><td>H(λ)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub></sub></td></tr><tr><td>B<sub>n</sub><sup>(1)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v+2-i-j</sub></sub></td></tr><tr><td>A<sub>2n-1</sub><sup>(2)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>i=1</sub><sup>n</sup> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Δ<sub>B<sub>n</sub><sup>(1)</sup></sub>(r)</td></tr><tr><td>C<sub>n</sub><sup>(1)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>i=1</sub><sup>n</sup> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v-i-j</sub></sub></td></tr><tr><td>D<sub>n+1</sub><sup>(2)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v+1-i-j</sub></sub></td></tr><tr><td>A<sub>2n</sub><sup>(2)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>i=1</sub><sup>n</sup> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Δ<sub>D<sub>n+1</sub><sup>(2)</sup></sub>(r)</td></tr><tr><td>A<sub>2n</sub><sup>(1)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v-i-j</sub></sub></td></tr><tr><td>D<sub>n</sub><sup>(1)</sup></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>i=1</sub><sup>n</sup> <sup>X<sub>2r<sub>i</sub></sub> X<sub>i</sub></sup>&frasl;<sub>X<sub>2i</sub> X<sub>r<sub>i</sub></sub></sub> Δ<sub>B<sub>n</sub><sup>(1)</sup></sub>(r)</td></tr></tbody></table>

**Remark 7.5.** The case $T = A_{n-1}^{(1)}$ in the above theorem is a reformulation of Theorem 1 in [DH11].

**Remark 7.6.** In the left-hand side products of the Theorem 7.4, one can note that if we take $(X_k)_{k \in \mathbb{Z}} = k$, the product of hook length is equal to 0 for any partition $\lambda \in \mathcal{P} \setminus C_T$. For instance, in type $A_{n-1}^{(1)}$, the product over all the hook lengths $(1 - n^2/h^2)$ of a partition $\lambda$ is equal to 0 if and only if $\lambda$ is not a $n$-core.

The integers $\alpha_T(i)$ and $\alpha'_T(i)$ can be expressed as a refinement of the cardinal of affine inversion sets.

The remaining steps in order to derive Nekrasov–Okounkov type identities is to consider Theorem 4.3 as an equality of multivariate power series setting $x_i = e^{-\epsilon_i}$ for $1 \le i \le n$. Then we specialize the set of variables $(x_i)$ so that we can use the Weyl $u$-dimension formula for, then we prove that the Nekrasov–Okounkov type identities hold for an infinite
**Theorem 7.4.** Let $(X_k)_{k \in \mathbb{Z}}$ be formal variables. For any affine type $T$ of rank $n$ in Table 1 and any element of the affine Grassmannian $c \in W_a^0$ such that $c = ut_\nu$ with $\nu \in Q \cap (-P_+) \text{ and } u \in W^\nu$. Set $\mathbf{r} = (r_i)_{1 \le i \le n}$ the vector such that $r_i = (-\tilde{\eta}^\vee \nu + u^{-1}(\mathring{\rho}) - \mathring{\rho})_i$ for all $i$. Let $\lambda \in C_T$ and $\mathcal{H}(\lambda)_T$ its corresponding elements in Table 1 and Table 2, respectively. Set $\alpha_T(i) = \#\{h_s \in \mathcal{H}(\lambda)_T, h_s = \tilde{\eta}^\vee - i\}$ and where $\alpha'_T(i) = \alpha_T(i) + \#\{h \in \bar{\lambda}, 2h = \tilde{\eta}^\vee - i\} + \#\{h \in \bar{\lambda}, h = \tilde{\eta}^\vee - i\}$. Then we have:

* if $T = A_{n-1}^{(1)}$

$$ (35) \qquad \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in C_n^{(1)} \cup D_{n+1}^{(2)} \cup A_{2n}^{(2)} \cup A'_{2n}^{(2)}$:

$$ (36) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h-\tilde{\eta}^{\vee}} X_{h-\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in B_n^{(1)} \cup A_{2n-1}^{(2)}$:

$$ (37) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

* if $T \in D_n^{(1)}$:

$$ (38) \qquad \prod_{h \in \bar{\lambda}} \frac{X_{2h+\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_{2h} X_h} \prod_{h \in \mathcal{H}(\lambda)_T} \frac{X_{h-\tilde{\eta}^{\vee}} X_{h+\tilde{\eta}^{\vee}}}{X_h^2} = \prod_{i=1}^{\tilde{\eta}^{\vee}-1} \left(\frac{X_{-i}}{X_i}\right)^{\alpha'_T(i)} \Delta_T(\mathbf{r}), $$

<table><caption>TABLE 2. Table of affine types with their corresponding hook length product</caption><thead><tr><th>T</th><th>H(λ)<sub>T</sub></th><th>Δ<sub>T</sub>(r)</th></tr></thead><tbody><tr><td>A<sup>(1)</sup><sub>n-1</sub></td><td>H(λ)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub></sub></td></tr><tr><td>B<sup>(1)</sup><sub>n</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v+2-i-j</sub></sub></td></tr><tr><td>A<sup>(2)</sup><sub>2n-1</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sup>n</sup><sub>i=1</sub> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Δ<sub>B<sup>(1)</sup><sub>n</sub></sub>(r)</td></tr><tr><td>C<sup>(1)</sup><sub>n</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sup>n</sup><sub>i=1</sub> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v-i-j</sub></sub></td></tr><tr><td>D<sup>(2)</sup><sub>n+1</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v+1-i-j</sub></sub></td></tr><tr><td>A<sup>(2)</sup><sub>2n</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sup>n</sup><sub>i=1</sub> <sup>X<sub>r<sub>i</sub></sub></sup>&frasl;<sub>X<sub>i</sub></sub> Δ<sub>D<sup>(2)</sup><sub>n+1</sub></sub>(r)</td></tr><tr><td>A<sup>(2)</sup><sub>2n</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sub>1≤i<j≤n</sub> <sup>X<sub>r<sub>i</sub>-r<sub>j</sub></sub> X<sub>r<sub>i</sub>+r<sub>j</sub></sub></sup>&frasl;<sub>X<sub>j-i</sub> X<sub>tilde{η}^v-i-j</sub></sub></td></tr><tr><td>D<sup>(1)</sup><sub>n</sub></td><td>H(λ<sup>tr</sup>)</td><td>Π<sup>n</sup><sub>i=1</sub> <sup>X<sub>2r<sub>i</sub></sub> X<sub>i</sub></sup>&frasl;<sub>X<sub>2i</sub> X<sub>r<sub>i</sub></sub></sub> Δ<sub>B<sup>(1)</sup><sub>n</sub></sub>(r)</td></tr></tbody></table>

**Remark 7.5.** The case $T = A_{n-1}^{(1)}$ in the above theorem is a reformulation of Theorem 1 in [DH11].

**Remark 7.6.** In the left-hand side products of the Theorem 7.4, one can note that if we take $(X_k)_{k \in \mathbb{Z}} = k$, the product of hook length is equal to 0 for any partition $\lambda \in \mathcal{P} \setminus C_T$. For instance, in type $A_{n-1}^{(1)}$, the product over all the hook lengths $(1 - n^2/h^2)$ of a partition $\lambda$ is equal to 0 if and only if $\lambda$ is not a $n$-core.

The integers $\alpha_T(i)$ and $\alpha'_T(i)$ can be expressed as a refinement of the cardinal of affine inversion sets.

The remaining steps in order to derive Nekrasov–Okounkov type identities is to consider Theorem 4.3 as an equality of multivariate power series setting $x_i = e^{-\epsilon_i}$ for $1 \le i \le n$. Then we specialize the set of variables $(x_i)$ so that we can use the Weyl $u$-dimension formula for, then we prove that the Nekrasov–Okounkov type identities hold for an infinite
number of ranks and conclude using a polynomiality argument. The Theorem 7.4 is the corner stone to bridge the gap between the Weyl formula for characters on the one hand and the hook length products on the other.

Let $u$ be a formal variable. The specializations of the indeterminates $x_i$'s as well as the corresponding hook length product and the specialization of the Laurent variables used in Theorem 7.4 are summarized in the following Table, where $\epsilon(c)$ is the signature of the corresponding Grassmannian element (see Table 1)

<table><caption>TABLE 3. Table of affine types with their corresponding specializations and hook length product</caption><thead><tr><th>T</th><th>(x<sub>i</sub>)<sub>1&le;i&le;n</sub></th><th>(X<sub>k</sub>)<sub>k&isin;Z</sub></th><th>hook length for the group character s<sub>-&eta;<sup>V</sup>&nu;+u<sup>-1</sup>(&rho;)-&rho;</sub></th></tr></thead><tbody><tr><td>A<sub>n-1</sub><sup>(1)</sup></td><td>(u<sup>i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c) &prod;<sub>h&isin;H(&lambda;)<sub>T</sub></sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h+&eta;<sup>V</sup></sup>)(1 - u<sup>h-&eta;<sup>V</sup></sup>)</div><div style="padding: 0 5px;">(1 - u<sup>h</sup>)<sup>2</sup></div></div></td></tr><tr><td>B<sub>n</sub><sup>(1)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)u<sup>-&eta;<sup>V</sup>&ell;(&lambda;&#x0303;)</td></tr><tr><td>A<sub>2n-1</sub><sup>(2)</sup></td><td>(u<sup>i</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c)(-1)<sup>&ell;(&lambda;&#x0303;)</sup>u<sup>-&eta;<sup>V</sup>/2&ell;(&lambda;&#x0303;)</td></tr><tr><td>C<sub>n</sub><sup>(1)</sup></td><td>(u<sup>i</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c)u<sup>&eta;<sup>V</sup>/2&ell;(&lambda;&#x0303;)</td></tr><tr><td>D<sub>n+1</sub><sup>(2)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)(-1)<sup>&lambda;&#x0303;</sup> &prod;<sub>h&isin;&lambda;&#x0303;</sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h-&eta;<sup>V</sup>/2</sup>)(1 - u<sup>h-&eta;<sup>V</sup></sup>)(1 + u<sup>h-&eta;<sup>V</sup></sup>/2)</div><div style="padding: 0 5px;">(1 - u<sup>2h</sup>)(1 - u<sup>h</sup>)(1 + u<sup>h-&eta;<sup>V</sup></sup>/2)</div></div> &prod;<sub>h&isin;H(&lambda;)<sub>T</sub></sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h-&eta;<sup>V</sup></sup>)(1 - u<sup>h-&eta;<sup>V</sup></sup>)</div><div style="padding: 0 5px;">(1 - u<sup>h</sup>)<sup>2</sup></div></div></td></tr><tr><td>A<sub>2n</sub><sup>(2)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)u<sup>&eta;<sup>V</sup>&ell;(&lambda;&#x0303;)</td></tr><tr><td>D<sub>n</sub><sup>(1)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)(-1)<sup>&ell;(&lambda;&#x0303;)</sup>u<sup>-2&eta;<sup>V</sup>&ell;(&lambda;&#x0303;)</td></tr></tbody></table>

To get the $u$-deformation of the Nekrasov-Okounkov formula as stated in (2) (and its generalizations in any affine classical type), we start with Theorem 4.3 and specialize each group character appearing in the left-hand side as prescribed in the second column of Table 3. Then, one gets the hook length formulas of the fourth column by using Theorem 7.4. Observe the signatures $\epsilon(c)$ of the affine Grassmannian elements simplify. In affine type A this yields (2). For the other types, we refer to [Wah23] § 5.3 for the precise formulas.

**Warning:** Theorem 4.3 uses the **dual** atomic length $L^V$. This subtlety imposes to switch each affine root system and its dual (obtained by transposing its generalized Cartan matrix) between Table 1 and Tables 2 and 3.

**Acknowledgements.** The authors would like to thank Olivier Brunat, Nathan Chapelier-Laget, and Thomas Gerber for their interest and their precious remarks as well as Philippe Nadeau for suggesting the reformulation of Theorem 7.4.

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number of ranks and conclude using a polynomiality argument. The Theorem 7.4 is the corner stone to bridge the gap between the Weyl formula for characters on the one hand and the hook length products on the other.

Let $u$ be a formal variable. The specializations of the indeterminates $x_i$'s as well as the corresponding hook length product and the specialization of the Laurent variables used in Theorem 7.4 are summarized in the following Table, where $\epsilon(c)$ is the signature of the corresponding Grassmannian element (see Table 1)

<table><caption>TABLE 3. Table of affine types with their corresponding specializations and hook length product</caption><thead><tr><th>T</th><th>(x<sub>i</sub>)<sub>1&le;i&le;n</sub></th><th>(X<sub>k</sub>)<sub>k&isin;Z</sub></th><th>hook length for the group character s<sub>-&eta;<sup>V</sup>&nu;+u<sup>-1</sup>(&rho;)-&rho;</sub></th></tr></thead><tbody><tr><td>A<sub>n-1</sub><sup>(1)</sup></td><td>(u<sup>i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c) &prod;<sub>h&isin;H(&lambda;)<sub>T</sub></sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h+&eta;<sup>V</sup></sup>)(1 - u<sup>h-&eta;<sup>V</sup></sup>)</div><div style="padding: 0 5px;">(1 - u<sup>h</sup>)<sup>2</sup></div></div></td></tr><tr><td>B<sub>n</sub><sup>(1)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)u<sup>-&eta;<sup>V</sup>&ell;(&lambda;&#773;)</td></tr><tr><td>A<sub>2n-1</sub><sup>(2)</sup></td><td>(u<sup>i</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c)(-1)<sup>&ell;(&lambda;&#773;)</sup>u<sup>-&eta;<sup>V</sup>/2&ell;(&lambda;&#773;)</td></tr><tr><td>C<sub>n</sub><sup>(1)</sup></td><td>(u<sup>i</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>k</sup></td><td>&epsilon;(c)u<sup>&eta;<sup>V</sup>/2&ell;(&lambda;&#773;)</td></tr><tr><td>D<sub>n+1</sub><sup>(2)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)(-1)<sup>&lambda;&#773;</sup> &prod;<sub>h&isin;&lambda;&#773;</sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h-&eta;<sup>V</sup>/2</sup>)(1 - u<sup>h-&eta;<sup>V</sup></sup>)(1 + u<sup>h+&eta;<sup>V</sup></sup>)/2</div><div style="padding: 0 5px;">(1 - u<sup>2h</sup>)(1 - u<sup>h</sup>)(1 + u<sup>h-&eta;<sup>V</sup></sup>/2)</div></div> &prod;<sub>h&isin;H(&lambda;)<sub>T</sub></sub> <div style="display: inline-block; vertical-align: middle; text-align: center;"><div style="border-bottom: 1px solid black; padding: 0 5px;">(1 - u<sup>h-&eta;<sup>V</sup></sup>)(1 - u<sup>h+&eta;<sup>V</sup></sup>)</div><div style="padding: 0 5px;">(1 - u<sup>h</sup>)<sup>2</sup></div></div></td></tr><tr><td>A<sub>2n</sub><sup>(2)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)u<sup>&eta;<sup>V</sup>&ell;(&lambda;&#773;)</td></tr><tr><td>D<sub>n</sub><sup>(1)</sup></td><td>(u<sup>2i-1</sup>)<sub>1&le;i&le;n</sub></td><td>X<sub>k</sub> = 1 - u<sup>2k</sup></td><td>&epsilon;(c)(-1)<sup>&ell;(&lambda;&#773;)</sup>u<sup>-2&eta;<sup>V</sup>&ell;(&lambda;&#773;)</td></tr></tbody></table>

To get the $u$-deformation of the Nekrasov-Okounkov formula as stated in (2) (and its generalizations in any affine classical type), we start with Theorem 4.3 and specialize each group character appearing in the left-hand side as prescribed in the second column of Table 3. Then, one gets the hook length formulas of the fourth column by using Theorem 7.4. Observe the signatures $\epsilon(c)$ of the affine Grassmannian elements simplify. In affine type A this yields (2). For the other types, we refer to [Wah23] § 5.3 for the precise formulas.

**Warning:** Theorem 4.3 uses the **dual** atomic length $L^V$. This subtlety imposes to switch each affine root system and its dual (obtained by transposing its generalized Cartan matrix) between Table 1 and Tables 2 and 3.

**Acknowledgements.** The authors would like to thank Olivier Brunat, Nathan Chapelier-Laget, and Thomas Gerber for their interest and their precious remarks as well as Philippe Nadeau for suggesting the reformulation of Theorem 7.4.

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CÉDRIC LECOUVEY: UNIV. DE TOURS, UMR CNRS 7013, INSTITUT DENIS POISSON, FRANCE

DAVID WAHICHE: UNIV. DE TOURS, UMR CNRS 7013, INSTITUT DENIS POISSON, FRANCE
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[NO06] N. Nekrasov and A. Okounkov, *Seiberg-Witten theory and random partitions*, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 525–596.

[Pé15] M. Pétréolle, *Quelques développements combinatoires autour des groupes de Coxeter et des partitions d'entiers*, Theses, Université Claude Bernard - Lyon I, November 2015.

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[RW18] E. Rains and S. Warnaar, *A Nekrasov-Okounkov formula for Macdonald polynomials*, J. Algebraic Combin. **48** (2018), no. 1, 1–30.

[Sta99] R. Stanley, *Enumerative combinatorics. Vol. 2*, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.

[STW23] E. Stucky, M. Thiel, and N. Williams, *Strange expectations in affine weyl groups*, arXiv 2309.14481, 2023.

[Wah23] D. Wahiche, *Some combinatorial interpretations of the macdonald identities for affine root systems and nekrasov-okounkov type formulas*, arXiv 2306.08071, 2023.

[Wes06] B. Westbury, *Universal characters from the Macdonald identities*, Adv. Math. **202** (2006), no. 1, 50–63.

CÉDRIC LECOUVEY: UNIV. DE TOURS, UMR CNRS 7013, INSTITUT DENIS POISSON, FRANCE

DAVID WAHICHE: UNIV. DE TOURS, UMR CNRS 7013, INSTITUT DENIS POISSON, FRANCE