About: Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $/wedge (V /oplus V^*)$ as well as its quotient $DR_W := /wedge (V /oplus V^*)/ /langle /wedge (V /oplus V^*)^{W}_+ /rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = /mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $/mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $/wedge (V /oplus V^*)$.   Goto Sponge  NotDistinct  Permalink

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  • Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $/wedge (V /oplus V^*)$ as well as its quotient $DR_W := /wedge (V /oplus V^*)/ /langle /wedge (V /oplus V^*)^{W}_+ /rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = /mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $/mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $/wedge (V /oplus V^*)$.
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  • Symmetry
  • Geometry
  • Lie groups
  • Theoretical chemistry
  • Group theory
  • Theoretical physics
  • Algebras
  • Representation theory
  • Multilinear algebra
  • Differential forms
  • Quotient objects
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