Defect 2 spin blocks of symmetric groups and canonical basis coefficients
Defect 2 spin blocks of symmetric groups and canonical basis coefficients
复制标题
对称群和规范基系数的缺陷 2 自旋块
DOI：
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
M. Fayers
中科院分区：
文献类型：
--
作者：
M. Fayers
本文研究了奇特征对称群的自旋表示的分解数问题。我们的主要目标是为缺陷块中的分解数找到一个组合公式<inline-formula content-type="math/mathml">
<p>This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2">
<mml:semantics>
<mml:mn>2</mml:mn>
<mml:annotation encoding="application/x-tex">2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, analogous to Richards’s formula for defect <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2">
<mml:semantics>
<mml:mn>2</mml:mn>
<mml:annotation encoding="application/x-tex">2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> blocks of symmetric groups.</p>
<p>By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding “<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-decomposition numbers”, i.e. the canonical basis coefficients in the level-<inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1">
<mml:semantics>
<mml:mn>1</mml:mn>
<mml:annotation encoding="application/x-tex">1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-deformed Fock space of type <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript 2 n Superscript left-parenthesis 2 right-parenthesis">
<mml:semantics>
<mml:msubsup>
<mml:mi>A</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn>2</mml:mn>
<mml:mi>n</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:msubsup>
<mml:annotation encoding="application/x-tex">A^{(2)}_{2n}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n plus 1">
<mml:semantics>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mi>n</mml:mi>
<mml:mo>+</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">2n+1</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. Along the way, we prove some general results on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q">
<mml:semantics>
<mml:mi>q</mml:mi>
<mml:annotation encoding="application/x-tex">q</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-decomposition numbers. This paper represents the first substantial progress on canonical bases in type <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript 2 n Superscript left-parenthesis 2 right-parenthesis">
<mml:semantics>
<mml:msubsup>
<mml:mi>A</mml:mi>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mn>2</mml:mn>
<mml:mi>n</mml:mi>
</mml:mrow>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mo stretchy="false">(</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
</mml:msubsup>
<mml:annotation encoding="application/x-tex">A^{(2)}_{2n}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>
舒伯特微积分和扭转爆炸
影响因子：
3.9
作者：
G. Williamson
通讯作者：
G. Williamson