On the graded quotients of the algebra of SL(2) characters of free groups

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} be a free group of rank 𝑛. An <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}(2) character of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} means the trace of an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}(2) representation of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} . Let 𝐾 be a field of characteristic 0. The automorphism group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{Aut}(F_{n}) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> F_{n} naturally acts on the commutative 𝐾-algebra of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}(2) characters. Then the augmentation ideal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>J</m:mi> <m:mi>n</m:mi> <m:mo>+</m:mo> </m:msubsup> </m:math> J_{n}^{+} of the commutative 𝐾-algebra generated by the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{SL}(2) characters is an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>F</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{Aut}(F_{n}) -invariant. The main purpose of this paper is to study the structure of the graded quotient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup>