Basis and dimension of the cyclotomic Hecke algebra

Exactly ${\mathrm{\ell }}^{d}d!$ spanning words. By [F1] the bounded ordered monomials span $H_d^{\mathrm{\Lambda }}$, and the number of tuples $(m_1,\dots ,m_d,w)$ with $0\le m_r<\mathrm{\ell }$ and $w\in S_d$ is ${\mathrm{\ell }}^{d}d!$. The spanning lemma is stated before, and independent of, the spectral/projector construction, and never imposes the first polynomial independently on every dot.

The dimension. By the sector isomorphism [F2] the algebra $H_d^{\mathrm{\Lambda }}$ is the direct sum of the finite family of sector algebras $H_{\alpha }^{\mathrm{\Lambda }}$ (one for each content $\alpha$ of height $d$), and each $H_{\alpha }^{\mathrm{\Lambda }}$ is isomorphic to the corresponding summand $R_{\alpha }^{\mathrm{\Lambda }}$ of the specialized KLR algebra. Summing dimensions, ${\dim }_F{H}_d^{\mathrm{\Lambda }}={\sum }_{\mathrm{ht}\alpha =d}{\dim }_F{H}_{\alpha }^{\mathrm{\Lambda }}={\sum }_{\mathrm{ht}\alpha =d}{\dim }_F{R}_{\alpha }^{\mathrm{\Lambda }}={\dim }_F{B}_d={\mathrm{\ell }}^{d}d!,$ the last equality being the total dimension statement of Cyclic type-A KLR regular Hom norm gives the total dimension; grouping the root components $R^{\mathrm{\Lambda }}(\beta )$ by their content $\alpha$ is legitimate because the idempotents of distinct contents are orthogonal. This is an integer identity, not an equality reduced modulo the characteristic.

Basis and boundary cases. A spanning family of ${\mathrm{\ell }}^{d}d!$ vectors in the ${\mathrm{\ell }}^{d}d!$-dimensional space $H_d^{\mathrm{\Lambda }}$ is a basis, so the bounded ordered monomials form an $F$-basis and ${\dim }_F{H}_d^{\mathrm{\Lambda }}={\mathrm{\ell }}^{d}d!$. For $d=0$ the singleton empty word gives $H_0^{\mathrm{\Lambda }}=F$ independently of $\mathrm{\ell }$; for $\mathrm{\ell }=0$ and $d>0$ the imposed relation is $1=0$, the algebra is zero and the empty family is its basis. No ambiguous $0^0$ convention is used. The late theorem does not feed the CRT or the early BK construction, and no Frobenius theorem or global highest-weight isometry is used.