Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras

Abstract

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level ℓ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce $\mathit{\text{S}}$-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagan's action in [17] by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.

Introduction

Kac-Moody algebras were independently introduced by Kac [10] and Moody [13]. Among them, affine Kac-Moody algebras have been particularly extensively studied for their beautiful representation theory as well as for their remarkable connections to other areas such as mathematical physics, number theory, combinatorics, and so on. Nevertheless, many basic questions are still unresolved. For instance the behaviour of weight multiplicities and combinatorial features of dominant maximal weights are not fully understood (see [12, Introduction]).

Throughout this paper, $\mathfrak{g}$ denotes an affine Kac-Moody algebra and $V(\mathrm{\Lambda })$ the irreducible highest weight module with highest weight $\mathrm{\Lambda }\in P^{+}$, where $P^{+}$ denotes the set of dominant integral weights. Due to Kac [11], all weights of $V(\mathrm{\Lambda })$ are given by the disjoint union of δ-strings attached to maximal weights and every maximal weight is conjugate to a unique dominant maximal weight under Weyl group action. So it would be quite natural to expect that better understanding of dominant maximal weights makes a considerable contribution towards the study of representation theory of affine Kac-Moody algebras.

In [11], Kac established lots of fundamental properties concerned with $\mathrm{wt}(\mathrm{\Lambda })$, the set of weights of $V(\mathrm{\Lambda })$, using the orthogonal projection $\overline{}:{\mathfrak{h}}^{⁎}\to {\mathfrak{h}}_0^{⁎}$. In particular, he showed that ${\max }^{+}(\mathrm{\Lambda })$, the set of dominant maximal weights, is in bijection with $\ell {\mathcal{C}}_{\mathrm{af}}\cap (\bar{\mathrm{\Lambda }}+\bar{Q})$ under this projection, thus it is finite. Here ℓ denotes the level of Λ. However, in the best knowledge of the authors, approachable combinatorial models, cardinality formulae and structure on ${\max }^{+}(\mathrm{\Lambda })$'s have not been available up to now except for limited cases, which motivates the present paper.

In 2014, Jayne and Misra [9] published noteworthy results about ${\max }^{+}(\mathrm{\Lambda })$ in $A_n^{(1)}$-case. They give an explicitly parametrization of ${\max }^{+}((\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i)$ in terms of paths for $0\le i\le n$ and $\ell \ge 2$, and present the following conjecture:

$$|{\max }^{+}(\ell {\mathrm{\Lambda }}_0)|=\frac{1}{(n+1)+\ell }\sum _{d|(n+1,\ell )}\phi (d)\left(\begin{array}{c}\hfill ((n+1)+\ell )/d\hfill \\ \hfill \ell /d\hfill \end{array}\right),$$

where φ is Euler's phi function. Notably this number gives the celebrated Catalan number when $\ell =n$. Soon after, this conjecture turned out to be affirmative in [22]. The proof therein largely depends on Sagan's congruence on q-binomial coefficients [17, Theorem 2.2].

The main purpose of this paper is to investigate ${\max }^{+}(\mathrm{\Lambda })$ by constructing bijections with several combinatorial models and a (bi)cyclic sieving phenomenon on the combinatorial models. As applications, we can obtain closed formulae of ${\max }^{+}(\mathrm{\Lambda })$ for all affine types, and observe interesting symmetries by considering ${\max }^{+}(\mathrm{\Lambda })$ for all ranks and levels.

Set

$$P_{\mathrm{cl}}^{+}:=P^{+}/\mathbb{Z}\mathrm{\delta }\text{ and }{P}_{\mathrm{cl},\ell }^{+}:=P_{\ell }^{+}/\mathbb{Z}\mathrm{\delta }\text{for }\ell \in {\mathbb{Z}}_{\ge 0},$$

where $P_{\ell }^{+}$ denotes the set of level ℓ dominant integral weights and δ denotes the canonical null root of $\mathfrak{g}$.

Given a nonnegative integer ℓ, we only consider classical dominant integral weights, that is, Λ in $P_{\mathrm{cl},\ell }^{+}$ because there is a natural bijection between ${\max }^{+}(\mathrm{\Lambda })$ and ${\max }^{+}(\mathrm{\Lambda }+k\delta )$ for every $k\in \mathbb{Z}$. We begin with the observation that the set $\ell {\mathcal{C}}_{\mathrm{af}}\cap (\bar{\mathrm{\Lambda }}+\bar{Q})$ can be embedded into $P_{\mathrm{cl},\ell }^{+}$ via the map

where Q denotes the root lattice, ${\varpi }_i:={\bar{\mathrm{\Lambda }}}_i$ and $m_0=\ell -{\sum }_{i=1}^n{a}_i^{\vee }{m}_i$.

We then define an equivalence relation ∼ on $P_{\mathrm{cl},\ell }^{+}$ by $\mathrm{\Lambda }\sim {\mathrm{\Lambda }}^{\prime }$ if and only if ${\iota }_{\mathrm{\Lambda }}={\iota }_{{\mathrm{\Lambda }}^{\prime }}$, equivalently $\ell {\mathcal{C}}_{\mathrm{af}}\cap (\bar{\mathrm{\Lambda }}+\bar{Q})=\ell {\mathcal{C}}_{\mathrm{af}}\cap (\overline{{\mathrm{\Lambda }}^{\prime }}+\bar{Q})$ (see Lemma 2.3). By definition, if $\mathrm{\Lambda }\sim {\mathrm{\Lambda }}^{\prime }$, then $|{\max }^{+}(\mathrm{\Lambda })|=|{\max }^{+}({\mathrm{\Lambda }}^{\prime })|$. Note that this equivalence relation is defined in [3] in a slightly different form. We should remark that, in [3], the authors mainly investigated a membership condition of weights for highest weight module $V(\mathrm{\Lambda })$ modulo a certain lattice, while we investigate $|{\max }^{+}(\mathrm{\Lambda })|$ and structures on the union of ${\max }^{+}(\mathrm{\Lambda })$'s.

Under the relation ∼, it turns out that the image of ${\iota }_{\mathrm{\Lambda }}$ coincides with the equivalence class of Λ. We provide a complete set of pairwise inequivalent representatives of the distinguished form $(\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i$, denoted by $\mathtt{DR}(P_{\mathrm{cl},\ell }^{+})$. For instances, in case where $\mathfrak{g}=A_n^{(1)}$, we have $\mathtt{DR}(P_{\mathrm{cl},\ell }^{+})=\{(\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i|0\le i\le n\}$ and in case where $\mathfrak{g}=E_6^{(1)}$, we have $\mathtt{DR}(P_{\mathrm{cl},\ell }^{+})=\{(\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i|i=0,1,6\}$ (see Table 2.2). It follows that

$$\underset{\mathrm{\Lambda }\in \mathtt{DR}(P_{\mathrm{cl},\ell }^{+})}{⨆}{P}_{\mathrm{cl},\ell }^{+}(\mathrm{\Lambda })=P_{\mathrm{cl},\ell }^{+},$$

where $P_{\mathrm{cl},\ell }^{+}(\mathrm{\Lambda })$ denotes the equivalence class of Λ under ∼. It should be noticed that $|P_{\mathrm{cl},\ell }^{+}(\mathrm{\Lambda })|=|{\max }^{+}(\mathrm{\Lambda })|$.

From this we derive a very significant consequence that the number of all equivalence classes is given by $\mathsf{N}:=[\bar{P}:\bar{Q}]$, where $\bar{P}/\bar{Q}$ is isomorphic to the fundamental group of the root system of ${\mathfrak{g}}_0$ except for $\mathfrak{g}=A_{2n}^{(2)}$ (see Table 2.1). Here ${\mathfrak{g}}_0$ denotes the subalgebra of $\mathfrak{g}$ which is of finite type.

Next, we introduce a new statistic ${\mathsf{ev}}_{{}_{\mathit{\text{S}}}}$, called the $\mathit{\text{S}}$-evaluation, on $P_{\mathrm{cl},\ell }^{+}$. Here $\mathit{\text{S}}$ is a certain set, called a root sieving set, which is characterized by a ${\mathbb{Z}}_{\hat{\mathsf{N}}}$-basis for the ${\mathbb{Z}}_{\mathsf{N}}$-kernel of the transpose of Cartan matrix associated ${\mathfrak{g}}_0$ (see (2.7) for the definition of $\hat{\mathsf{N}}$, and Lemma 2.12 and Convention 2.13 for details). In more detail, for all affine Kac-Moody algebras except for $D_n^{(1)}(n{\equiv }_{2}0)$, $\mathit{\text{S}}$ consists of a single element $(s_1,\dots ,s_n)$ and

$${\mathsf{ev}}_{{}_{\mathit{\text{S}}}}\left(\sum _{0\le i\le n}{m}_i{\mathrm{\Lambda }}_i\right):=\sum _{1\le i\le n}{s}_i{m}_i\text{ for }\mathrm{\Lambda }=\sum _{0\le i\le n}{m}_i{\mathrm{\Lambda }}_i.$$

In case where $\mathfrak{g}=D_n^{(1)}(n{\equiv }_{2}0)$, we have $\mathit{\text{S}}=\{{\mathit{\text{s}}}^{(1)}=(0,0,\dots ,0,2,2),{\mathit{\text{s}}}^{(2)}=(2,0,2,0,\dots ,2,0,2,0)\}$. For the $\mathit{\text{S}}$-evaluation of this type, see (2.10). Finally, exploiting this statistic, we characterize the equivalence class of $\mathrm{\Lambda }\in \mathtt{DR}(P_{\mathrm{cl},\ell }^{+})$ in terms of $\mathit{\text{S}}$-evaluation (Theorem 2.14).

Quite interestingly, the $\mathit{\text{S}}$-evaluation on $P_{\mathrm{cl},\ell }^{+}$ leads us to construct a (bi)cyclic sieving phenomenon on it. The cyclic sieving phenomenon, introduced by Reiner-Stanton-White in [14], is generalized and developed in various aspects including combinatorics and representation theory (see [1], [2], [5], [16], [18] for examples).

Let us briefly recall the cyclic sieving phenomenon. Let X be a finite set, with an action of a cyclic group C of order m, and $X(q)$ a polynomial in q with nonnegative integer coefficients. For $d\in {\mathbb{Z}}_{>0}$, let ${\omega }_d$ be a dth primitive root of the unity. We say that $(X,C,X(q))$ exhibits the cyclic sieving phenomenon if, for all $g\in C$, we have $|X^g|=X({\omega }_{\mathrm{o}(g)})$, where $\mathrm{o}(g)$ is the order of g and $X^g$ is the fixed point set under the action of g.

Let us explain our initial motivation. It was shown in [14, Theorem 1.1] that $(\left(\begin{array}{c}\hfill [0,n]\hfill \\ \hfill \ell \hfill \end{array}\right),C_{n+1},{\left[\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right]}_q)$ exhibits the cyclic sieving phenomenon. Here $\left(\begin{array}{c}\hfill [0,n]\hfill \\ \hfill \ell \hfill \end{array}\right)$ denotes the set of all ℓ-multisets on $\{0,1,\dots ,n\}$, $C_{n+1}$ a fixed cyclic group of order $n+1$, and ${\left[\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right]}_q$ the q-binomial coefficient of $\left(\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right)$. We identify $\left(\begin{array}{c}\hfill [0,n]\hfill \\ \hfill \ell \hfill \end{array}\right)$ with $P_{\mathrm{cl},\ell }^{+}$ in $A_n^{(1)}$-type as $C_{n+1}$-sets and let

$$P_{\mathrm{cl},\ell }^{+}(q):={\left[\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right]}_q.$$

Then we observe that the generating function of $P_{\mathrm{cl},\ell }^{+}(q)$ $(\ell \ge 0)$ can be expressed in terms of the root sieving set $\mathit{\text{S}}=\{(s_1,s_2,\dots ,s_n)=(1,2,\dots ,n)\}$ and the canonical center $c=h_0+h_1+h_2+\dots +h_n={\sum }_{i=0}^n{a}_i^{\vee }{h}_i$ as follows:

$$\sum _{\ell \ge 0}{P}_{\mathrm{cl},\ell }^{+}(q)t^{\ell }:=\sum _{\ell \ge 0}{\left[\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right]}_q{t}^{\ell }=\prod _{0\le i\le n}\frac{1}{1-q^i{t}^1}=\prod _{0\le i\le n}\frac{1}{1-q^{s_i}{t}^{a_i^{\vee }}},$$

where $s_0$ is set to be 0. From this product identity it follows that $P_{\mathrm{cl},\ell }^{+}(q)={\sum }_{\mathrm{\Lambda }\in P_{\mathrm{cl},\ell }^{+}}{q}^{{\mathsf{ev}}_{{}_{\mathit{\text{S}}}}(\mathrm{\Lambda })}$. Furthermore, since $C_{n+1}$ is isomorphic to $\bar{P}/\bar{Q}$, we conclude that the triple $(P_{\mathrm{cl},\ell }^{+},\bar{P}/\bar{Q},P_{\mathrm{cl},\ell }^{+}(q))$ also exhibits the cyclic sieving phenomenon.

Then it is natural to ask whether there exists a triple for other affine Kac-Moody algebras exhibiting the cyclic sieving phenomenon or not. Canonically, one can construct the triple in uniform way for all affine Kac-Moody algebras as follows: We first take $P_{\mathrm{cl},\ell }^{+}$ as the underlying set. Second, writing the canonical center as $c={\sum }_{i=0}^n{a}_i^{\vee }{h}_i$, we take $P_{\mathrm{cl},\ell }^{+}(q)$ from the following geometric series (by mimicking the $A_n^{(1)}$-case):

$$\{\begin{array}{cc}{\sum }_{\ell \ge 0}{P}_{\mathrm{cl},\ell }^{+}(q)t^{\ell }:=\prod _{0\le i\le n}\frac{1}{1-q^{s_i}{t}^{a_i^{\vee }}},\hfill & \text{ if }\mathfrak{g}\text{ is not of type }{D}_n^{(1)}\text{ for even }\mathit{\text{n}},\hfill \\ {\sum }_{\ell \ge 0}{P}_{\mathrm{cl},\ell }^{+}(q_1,q_2)t^{\ell }:=\prod _{0\le i\le n}\frac{1}{1-q_1^{{\mathfrak{s}}_i^{(1)}}{q}_2^{{\mathfrak{s}}_i^{(2)}}{t}^{a_i^{\vee }}}\hfill & \text{ if }\mathfrak{g}\text{ is of type }{D}_n^{(1)}\text{ for even }\mathit{\text{n}},\hfill \end{array}$$

where $s_0$ is set to be 0 (see (4.2) and (5.2)). Then we have

$$\{\begin{array}{cc}{P}_{\mathrm{cl},\ell }^{+}(q)={\sum }_{\mathrm{\Lambda }\in P_{\mathrm{cl},\ell }^{+}}{q}^{{\mathsf{ev}}_{{}_{\mathit{\text{S}}}}(\mathrm{\Lambda })}\hfill & \text{ if }\mathfrak{g}\text{ is not of type }{D}_n^{(1)}\text{ for even }\mathit{\text{n}},\hfill \\ P_{\mathrm{cl},\ell }^{+}(q_1,q_2)={\sum }_{\mathrm{\Lambda }\in P_{\mathrm{cl},\ell }^{+}}{q}_1^{{\mathsf{ev}}_{{\mathfrak{s}}^{(1)}}(\mathrm{\Lambda })}{q}_2^{{\mathsf{ev}}_{{\mathfrak{s}}^{(2)}}(\mathrm{\Lambda })}\hfill & \text{ if }\mathfrak{g}\text{ is of type }{D}_n^{(1)}\text{ for even }\mathit{\text{n}}.\hfill \end{array}$$

Finally, take $\bar{P}/\bar{Q}$ as the (bi)cyclic group, which completes the triple:

$$(P_{\mathrm{cl},\ell }^{+},\bar{P}/\bar{Q},P_{\mathrm{cl},\ell }^{+}(q))\text{ (resp. }(P_{\mathrm{cl},\ell }^{+},\bar{P}/\bar{Q},P_{\mathrm{cl},\ell }^{+}(q_1,q_2))).$$

We assign an appropriate $\bar{P}/\bar{Q}$-action on $P_{\mathrm{cl},\ell }^{+}$ (see (4.10) and (5.1)), and prove that the triple exhibits the (bi)cyclic sieving phenomenon, which can be understood as a natural generalization of the cyclic sieving triple $(\left(\begin{array}{c}\hfill [0,n]\hfill \\ \hfill \ell \hfill \end{array}\right),C_{n+1},{\left[\begin{array}{c}\hfill n+\ell \hfill \\ \hfill \ell \hfill \end{array}\right]}_q)$ in aspect of affine Kac-Moody algebras.

For the proof, we employ the following strategy. For each divisor d of $\mathsf{N}$, we introduce a set ${\mathbf{M}}_{\ell }(rd,d;\mathit{\nu },{\mathit{\nu }}^{\prime })$ equipped with a $C_d$-action obtained by generalizing Sagan's action on $(0,1)$-words in [17]. Here, $r,\mathit{\nu },{\mathit{\nu }}^{\prime }$ are chosen so that ${\mathbf{M}}_{\ell }(rd,d;\mathit{\nu },{\mathit{\nu }}^{\prime })$ can be identified with $P_{\mathrm{cl},\ell }^{+}$ by permuting indices properly. Then we show that $|{\mathbf{M}}_{\ell }{(rd,d;\mathit{\nu },{\mathit{\nu }}^{\prime })}^{C_d}|=\left|{\left(P_{\mathrm{cl},\ell }^{+}\right)}^g\right|$ for all $g\in \bar{P}/\bar{Q}$ of order d. We end the proof by showing

$$|{\mathbf{M}}_{\ell }{(rd,d;\mathit{\nu },{\mathit{\nu }}^{\prime })}^{C_d}|=P_{\mathrm{cl},\ell }^{+}({\zeta }_{\mathsf{N}}^{\mathsf{N}/d}).$$

From the above sieving phenomena, we derive closed formulae for $|{\max }^{+}(\mathrm{\Lambda })|$ for all $\mathrm{\Lambda }\in P_{\mathrm{cl},\ell }^{+}$ and for affine Kac-Moody algebras of arbitrary type. For the classical types, they are explicitly written as a sum of binomial coefficients (see Section 6.1). For instance, in case where $A_n^{(1)}$ type, we obtain

$$|{\max }^{+}((\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i)|=\sum _{d|(n+1,\ell ,i)}\frac{d}{(n+1)+\ell }\sum _{d^{\prime }|(\frac{n+1}{d},\frac{\ell }{d})}\mu (d^{\prime })\left(\begin{array}{c}\hfill ((n+1)+\ell )/d{d}^{\prime }\hfill \\ \hfill \ell /d{d}^{\prime }\hfill \end{array}\right),$$

which is a vast generalization of (0.1) (see also Theorem 4.6).

Let us view ${\{|{\max }^{+}(\mathrm{\Lambda })|\}}_{n,\ell }$ as a sequence expressed in terms of n and ℓ. Exploiting our closed formulae, we can also derive recursive formulae for $|{\max }^{+}(\mathrm{\Lambda })|$ (except for type $A_n^{(1)}$) and their corresponding triangular arrays. It is quite interesting to observe that several triangular arrays are already known in different contexts. For example, when $\mathfrak{g}$ is of affine C-type, our triangular arrays are known as Lozanić's triangle and its Pascal complement (see Subsection 6.2.1). Also, the triangular array for twisted affine even A-type is Pascal triangle with duplicated diagonals (see Appendix A).

Going further, we observe interesting interrelations among the triangular arrays of various affine Kac-Moody algebras (see Appendix A). Surprisingly, all triangular arrays for classical affine type except for untwisted affine C-type can be constructed by boundary conditions and the triangular array of twisted affine even A-type. Similarly, the triangular arrays for untwisted affine C-type can be constructed by boundary conditions and Pascal triangle. Considering that the triangular array of twisted affine even A-type can be obtained from Pascal triangle, we can conclude that all triangular arrays for classical affine types can be obtained from boundary conditions and Pascal triangle only.

As another byproduct of our closed formulae, we observe a symmetry which appears as level and rank are switched in a certain way. For instance, if $(n+1,\ell ,i)=(\ell ,n+1,j)$ for some $0\le i\le n$ and $0\le j\le \ell -1$, then

$$|{\max }_{A_n^{(1)}}^{+}((\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_i)|=|{\max }_{A_{\ell -1}^{(1)}}^{+}(n{\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_j)|.$$

This symmetry is compatible with the classical level-rank duality for $A_n^{(1)}$ studied by Frenkel in [6] (see Subsection 6.2.2). With the closed formulae of ${\max }^{+}(\mathrm{\Lambda })$ in terms of binomial coefficients, we can observe interesting symmetries for all classical affine types. For instances, we have

$$\{\begin{array}{cc}|{\max }_{B_n^{(1)}}^{+}(\ell {\mathrm{\Lambda }}_0)|=|{\max }_{B_{(\ell -1)/2}^{(1)}}^{+}((2n+1){\mathrm{\Lambda }}_0)|,\hfill & \text{ if}\mathit{\text{ℓ}}\text{is odd},\hfill \\ |{\max }_{B_n^{(1)}}^{+}((\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_n)|=|{\max }_{B_{\ell /2-1}^{(1)}}^{+}((2n+1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_{\ell /2-1})|\hfill & \text{ if}\mathit{\text{ℓ}}\text{is even},\hfill \end{array}$$

by exchanging n with $(\ell -1)/2$, and n with $\ell /2-1$, respectively, since

$$|{\max }_{B_n^{(1)}}^{+}(\ell {\mathrm{\Lambda }}_0)|=\left(\begin{array}{c}n+\lfloor \frac{\ell }{2}\rfloor \\ n\end{array}\right)+\left(\begin{array}{c}n+\lfloor \frac{\ell -1}{2}\rfloor \\ n\end{array}\right),|{\max }_{B_n^{(1)}}^{+}((\ell -1){\mathrm{\Lambda }}_0+{\mathrm{\Lambda }}_n)|=\left(\begin{array}{c}n+\lfloor \frac{\ell -1}{2}\rfloor \\ n\end{array}\right)+\left(\begin{array}{c}n+\lfloor \frac{\ell }{2}\rfloor -1\\ n\end{array}\right).$$

This paper is organized as follows. In Section 1, we introduce necessary notations and backgrounds for affine Kac-Moody algebras, highest weight modules and classical results on dominant maximal weights. In Section 2, we define an equivalence relation ∼ on $P_{\mathrm{cl},\ell }^{+}$ satisfying that the equivalence class of $\mathrm{\Lambda }\in P_{\mathrm{cl},\ell }^{+}$ has the same cardinality with ${\max }^{+}(\mathrm{\Lambda })$. Then we provide the set $\mathtt{DR}(P_{\mathrm{cl},\ell }^{+})$ of distinguished representatives, and characterize all equivalence classes in terms of $\mathit{\text{S}}$-evaluation with our sieving set $\mathit{\text{S}}$. In Section 3, we generalize Sagan's action with consideration on the result in Section 2 and prove that the generalized action gives cyclic action on $P_{\mathrm{cl},\ell }^{+}$ indeed. In Section 4, we prove that our triple for affine Kac-Moody algebras except $D_n^{(1)}$ for even n exhibits the cyclic sieving phenomenon. In Section 5, we prove the triple for $D_n^{(1)}$ for even n exhibits bicyclic sieving phenomenon. In Section 6, we derive closed formulae, recursive formulae, and level-rank duality for the sets of dominant maximal weights from the cyclic sieving phenomenon. In Appendix A and B, we list all triangular arrays and level-rank duality for affine Kac-Moody algebras, not dealt with in Section 6.