<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\in \mathbb {Z}^{{\ge }2}, \ell \in \mathbb {Z}^{{\ge }1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>≥</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>ℓ</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>≥</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. In this paper we use (some slightly modified versions of) the distinguished bases <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{\mathcal {B}_{\mathfrak {s}\mathfrak {t}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="script">B</mi> <mrow> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">t</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{\check{\mathcal {B}}_{\mathfrak {s}\mathfrak {t}}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msub> <mover accent="true"> <mi mathvariant="script">B</mi> <mo stretchy="false">ˇ</mo> </mover> <mrow> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">t</mi> </mrow> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of the cyclotomic Hecke algebra <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {H}_{\ell ,n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> of type <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$G(\ell ,1,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ℓ</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> introduced by Mathas and the first named author (Hu and Mathas, A. Math. Ann. <b>364</b>, 1189–1254, <CitationRef CitationID="CR16">2016</CitationRef>) to study the alternating cyclotomic Hecke algebra <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathscr {H}_{\ell ,n}^{\#}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>#</mo> </msubsup> </math></EquationSource> </InlineEquation>. We construct an explicit integral basis for the alternating cyclotomic Hecke algebra <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathscr {H}_{\ell ,n}^{\#}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>n</mi> </mrow> <mo>#</mo> </msubsup> </math></EquationSource> </InlineEquation> of arbitrary higher levels. We also present an explicit seminormal basis for the semisimple alternating cyclotomic Hecke algebra. We show that the alternating cyclotomic Hecke algebra is a symmetric algebra over an infinite field.</p>

Bases and Symmetric Structure Of Alternating Cyclotomic Hecke Algebras

期刊论文
发表日期：

2026年2月16日

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Jun Hu,
Xiaolin Shi,
Shixuan Wang

摘要

Let $n\in \mathbb {Z}^{{\ge }2}, \ell \in \mathbb {Z}^{{\ge }1}$ $n \in Z^{\geq 2}, ℓ \in Z^{\geq 1}$ . In this paper we use (some slightly modified versions of) the distinguished bases $\{\mathcal {B}_{\mathfrak {s}\mathfrak {t}}\}$ ${B_{s t}}$ and $\{\check{\mathcal {B}}_{\mathfrak {s}\mathfrak {t}}\}$ ${{\overset{ˇ}{B}}_{s t}}$ of the cyclotomic Hecke algebra $\mathscr {H}_{\ell ,n}$ $H_{ℓ, n}$ of type $G(\ell ,1,n)$ $G (ℓ, 1, n)$ introduced by Mathas and the first named author (Hu and Mathas, A. Math. Ann. 364, 1189–1254, 2016) to study the alternating cyclotomic Hecke algebra $\mathscr {H}_{\ell ,n}^{\#}$ $H_{ℓ, n}^{#}$ . We construct an explicit integral basis for the alternating cyclotomic Hecke algebra $\mathscr {H}_{\ell ,n}^{\#}$ $H_{ℓ, n}^{#}$ of arbitrary higher levels. We also present an explicit seminormal basis for the semisimple alternating cyclotomic Hecke algebra. We show that the alternating cyclotomic Hecke algebra is a symmetric algebra over an infinite field.