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Jules Lamers
Jules Lamers
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Yes, these exist and are known as Young's orthogonal matrix units.

To construct $E_{T,T'}$ note that $e_T \,\mathbb{C}[S_n] e_{T'}$$E_T \,\mathbb{C}[S_n] \, E_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S_n]$ such that $E_T \, x\, E_{T'}$ is nonzero, for example $x \in S_n$ the unique permutation that sends $T'$ to $T$. (One can also use the Baxterised elements to exchange boxes labelled $i,i+1$ in the tableaux successively to get from $T'$ to $T$.)

For the $q$-case (Hecke algebra) their construction is given in Prop 2.1 of Ram and Wenzl, "Matrix units for centralizer algebras", J Algebra 145 (1992) 378-395.

Yes, these exist and are known as Young's orthogonal matrix units.

To construct $E_{T,T'}$ note that $e_T \,\mathbb{C}[S_n] e_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S_n]$ such that $E_T \, x\, E_{T'}$ is nonzero, for example $x \in S_n$ the unique permutation that sends $T'$ to $T$. (One can also use the Baxterised elements to exchange boxes $i,i+1$ successively to get from $T'$ to $T$.)

For the $q$-case (Hecke algebra) their construction is given in Prop 2.1 of Ram and Wenzl, "Matrix units for centralizer algebras", J Algebra 145 (1992) 378-395.

Yes, these exist and are known as Young's orthogonal matrix units.

To construct $E_{T,T'}$ note that $E_T \,\mathbb{C}[S_n] \, E_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S_n]$ such that $E_T \, x\, E_{T'}$ is nonzero, for example $x \in S_n$ the unique permutation that sends $T'$ to $T$. (One can also use the Baxterised elements to exchange boxes labelled $i,i+1$ in the tableaux successively to get from $T'$ to $T$.)

For the $q$-case (Hecke algebra) their construction is given in Prop 2.1 of Ram and Wenzl, "Matrix units for centralizer algebras", J Algebra 145 (1992) 378-395.

Source Link
Jules Lamers
Jules Lamers
  • 2k
  • 1
  • 18
  • 35

Yes, these exist and are known as Young's orthogonal matrix units.

To construct $E_{T,T'}$ note that $e_T \,\mathbb{C}[S_n] e_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S_n]$ such that $E_T \, x\, E_{T'}$ is nonzero, for example $x \in S_n$ the unique permutation that sends $T'$ to $T$. (One can also use the Baxterised elements to exchange boxes $i,i+1$ successively to get from $T'$ to $T$.)

For the $q$-case (Hecke algebra) their construction is given in Prop 2.1 of Ram and Wenzl, "Matrix units for centralizer algebras", J Algebra 145 (1992) 378-395.