Murnaghan-Nakayama rule for the cyclotomic Hecke algebra and applications

Abstract

We derive a Murnaghan-Nakayama rule for irreducible characters of the cyclotomic Hecke algebra on certain standard elements, which fully determine their values. This work builds upon our recent multi-parameter Murnaghan-Nakayama rule for Macdonald polynomials. Our Murnaghan-Nakayama rule can be readily specialized to retrieve various existing rules, including those for the complex reflection group of type $G(m, 1, n)$ and the Iwahori-Hecke algebra in types $A$ and $B$. In a dual picture, we establish an iterative formula for the irreducible characters on upper multipartitions, utilizing the vertex operator realization of Schur functions. As applications we derive an Regev-type formula and an Lübeck-Prasad-Adin-Roichman-type formula for the cyclotomic Hecke algebra, thereby extending the corresponding formulas for the Iwahori-Hecke algebra in type $A$ and the complex reflection group of type $G(m,1,n)$ to the setting of the cyclotomic Hecke algebra, respectively. Finally, we introduce the notion of the multiple bitrace of the cyclotomic Hecke algebra to formulate the second orthogonal relation of the irreducible characters. We also provide a general combinatorial rule to compute the multiple bitrace.

Figures (7)

• Figure 1: $\lambda$ is obtained from $\mu$ by adding a horizontal strip (to get $\nu$) and then adding a vertical strip (to $\nu$).
• Figure 2: ${\rm SE}(\bm{\lambda}/\bm{\mu})$ consists of the green boxes. The subset formed by the boxes with bullets is an element of $\mathcal{R}_9(\bm{\lambda}/\bm{\mu})$.
• Figure 3: An example for $R_{\bm{\alpha},\bm{\beta}}$ where $\bm{\alpha}=((2,3), (3,2,2))$ and $\bm{\beta}=((2,2),(3))$. The corresponding skew character can be written as $\chi^{(\bm{\alpha},\bm{\beta})}=\chi^{(2)}\hat{\otimes}\chi^{(3)}\hat{\otimes}\chi^{(1^2)}\hat{\otimes}\chi^{(1^2)}\hat{\otimes}\chi^{(3)}\hat{\otimes}\chi^{(2)}\hat{\otimes}\chi^{(2)}\hat{\otimes}\chi^{(1^3)}.$
• Figure 4: Four possible removal processes for $\eta^{(3,1),(2,1,1),(2,2)}_{(2,1);r}$. Here $t=1$ and we choose $\sigma=(1)$. The sum terms they contribute are respectively $u^r_2(1-q^{-2})^3(-q^{-2})^0$, $u^r_2(1-q^{-2})^3(-q^{-2})^0$, $u^r_2(1-q^{-2})^2(-q^{-2})^0$ and $u^r_2(1-q^{-2})^2(-q^{-2})^0$.
• Figure 5: $\lambda=(7,7,7,6,3,3,2,1)$, the boundary of $\lambda$ is marked by arrows. Thus, $\partial(\lambda)=10101001 \mid 1101000$ and the anchor of $\partial(\lambda)$ is $8$.

Theorems & Definitions (63)

• Theorem 2.1
• Lemma 3.1
• proof
• Example 3.2
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Example 3.5
• Example 3.6