Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Swanson, Joshua P.; Wallach, Nolan R.

doi:10.1016/j.jcta.2021.105474

Cited by 12 publications

(16 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus I, f = (τ (I), f ) = (I * , f ), so H(V * ) is the orthogonal complement of I * under a positive-definite Hermitian form. Now (27) follows from Lemma 2.10 in this case.…”

Section: Harmonics and Semi-invariant Basesmentioning

confidence: 82%

“…It is well-known that the top-degree component of S(V * )/I * is the image of S(V * ) det V , which has motivated much of our work. We will explore which bidegrees of S(V * ) ⊗ ∧M * /J * M are non-zero in a future article [27].…”

Section: Let J *mentioning

confidence: 99%

“…As for (27), first suppose F ⊂ C is closed under complex conjugation. By Lemma 2.11, we have τ (S(V ) G ) = S(V * ) G and τ (I) = I * .…”

Section: Harmonics and Semi-invariant Basesmentioning

confidence: 99%

See 2 more Smart Citations

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Swanson¹,

Wallach²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "topdown" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon [22], Orlik-Solomon [14], and Shepler [20,21] describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki [29] which provides a representation theoretic-model for the Delta conjecture of Haglund-Remmel-Wilson [7] in terms of a certain non-commutative coinvariant algebra for the symmetric group. In particular, we verify the alternating component of a specialization of Zabrocki's conjecture.

show abstract

“…Thus I, f = (τ (I), f ) = (I * , f ), so H(V * ) is the orthogonal complement of I * under a positive-definite Hermitian form. Now (27) follows from Lemma 2.10 in this case.…”

Section: Harmonics and Semi-invariant Basesmentioning

confidence: 82%

Section: Let J *mentioning

confidence: 99%

See 1 more Smart Citation

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Swanson¹,

Wallach²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The symmetric function expressions and representation theoretic interpretation was extended further to include the quotient of two sets of commuting and two sets of anticommuting variables in [8] to what is known as the Theta Conjecture. At present, this also remains an open conjecture, but progress has been made on some special cases [20,21,28,29].…”

Section: Introductionmentioning

confidence: 99%

Quasisymmetric harmonics of the exterior algebra

Bergeron¹,

Chan²,

Solomon³

et al. 2022

Preprint

View full text Add to dashboard Cite

We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let Rn denote the polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables:(1) The quasisymmetric polynomials in Rn form a commutative sub-algebra of Rn.(2) There is a basis of the quotient of Rn by the ideal In generated by the quasisymmetric polynomials in Rn that is indexed by ballot sequences. The Hilbert series of the quotient is given bywhere f (n−k,k) is the number of standard tableaux of shape (n − k, k).(3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition

show abstract

“…The ring DR n is a bigraded S n -module; Haiman used algebraic geometry to calculate its isomorphism type [10]. In recent years, researchers in algebraic combinatorics studied variants of DR n involving mixtures of commuting and anticommuting variables [1,2,3,9,11,12,13,14,16,17,18,19]. Drawing terminology from supersymmetry, we will refer to commuting variables as bosonic and anticommuting variables as fermionic.…”

Section: Introductionmentioning

confidence: 99%

A proof of the fermionic Theta coinvariant conjecture

Iraci¹,

Rhoades²,

Romero³

2022

Preprint

View full text Add to dashboard Cite

Let (x1, . . . , xn, y1, . . . , yn) be a list of 2n commuting variables, (θ1, . . . , θn, ξ1, . . . , ξn) be a list of 2n anticommuting variables, and C[xn, yn] ⊗ ∧{θn, ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded Snisomorphism type of C[xn, yn] ⊗ ∧{θn, ξ n }/I where I is the ideal generated by Sn-invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables equal xi, yi equal to zero.

show abstract

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Cited by 12 publications

References 23 publications

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

Quasisymmetric harmonics of the exterior algebra

A proof of the fermionic Theta coinvariant conjecture

Contact Info

Product

Resources

About