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- W3173957034 abstract "Abstract This paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formula can be written as an explicit non-negative linear combination of modules $$[E^{varvec{nu }}_{f,e}]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , which are the modular reductions of simple modules of related Hecke algebras in characteristic zero. The coefficient of $$[E^{varvec{nu }}_{f,e}]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mo>[</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and by the characteristic of the field. As a consequence we give an explicit upper bound for the decomposition numbers in characteristic $$p>0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in terms of linear combinations of decomposition numbers of a cyclotomic Hecke algebra at $$ep^r$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>e</mml:mi> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> </mml:math> th roots of unity in characteristic zero, for $$rge 0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Finally, we prove a new and more elegant “classical” Jantzen sum formula for these algebras." @default.
- W3173957034 created "2021-07-05" @default.
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- W3173957034 date "2022-03-02" @default.
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- W3173957034 title "Positive Jantzen sum formulas for cyclotomic Hecke algebras" @default.
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