Multiplicity-free induced characters of symmetric groups

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a non-negative integer. Combining algebraic and combinatorial techniques, we investigate for which pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G,\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a subgroup <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the symmetric group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an irreducible character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ</mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the induced character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho up-arrow Superscript upper S Super Subscript n Baseline"> <mml:semantics> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mspace width="negativethinmathspace"/> <mml:msup> <mml:mo stretchy="false">↑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\rho \!\uparrow ^{S_n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is multiplicity-free. As a result, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 66"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>66</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 66</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we classify all subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G less-than-or-equal-to upper S Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G\leq S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which give rise to such a pair. Moreover, for the majority of these groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we identify all the possible choices of the irreducible character <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ</mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, assuming <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 73"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>73</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\geq 73</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• arXiv (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View