A proof of the fermionic Theta coinvariant conjecture

Iraci, Alessandro; Rhoades, Brendon; Romero, Marino

doi:10.48550/arxiv.2202.04170

Cited by 3 publications

(5 citation statements)

References 18 publications

(44 reference statements)

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“…Because the ideal J n is principal we can understand this quotient with much more detail. This quotient ring is a special case of the ring recently studied in [22,23].…”

Section: The Ideal Generated By Symmetric Invariantsmentioning

confidence: 99%

“…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 [22,23,30,31].…”

Section: Introductionmentioning

confidence: 98%

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Quasisymmetric harmonics of the exterior algebra

Bergeron¹,

Chan²,

Soltani³

et al. 2023

Can. Math. Bull.

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We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let R n denote the ring of 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 R n form a commutative sub-algebra of R n .(2) There is a basis of the quotient of R n by the ideal I n generated by the quasisymmetric polynomials in R n that is indexed by ballot sequences. The Hilbert series of the quotient is given by 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

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“…Because the ideal J n is principal we can understand this quotient with much more detail. This quotient ring is a special case of the ring recently studied in [22,23].…”

Section: The Ideal Generated By Symmetric Invariantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Quasisymmetric harmonics of the exterior algebra

Bergeron¹,

Chan²,

Soltani³

et al. 2023

Can. Math. Bull.

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show abstract

“…Because the ideal J n is prime we can understand this quotient with much more detail. This quotient ring is a special case of the ring recently studied in [20,21].…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Quasisymmetric harmonics of the exterior algebra

Bergeron¹,

Chan²,

Solomon³

et al. 2022

Preprint

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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

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“…We should mention that it was also conjectured in [10] that Theta operators give the graded Frobenius characteristic of the S n coinvariants in two sets of commuting variables and two sets of anticommuting variables. The purely fermionic case has recently been proved in [24].…”

Section: Introductionmentioning

confidence: 99%

Delta and Theta Operator Expansions

Iraci¹,

Romero²

2022

Preprint

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We give an elementary symmetric function expansion for M ∆m γ e 1 Πe * λ and M ∆m γ e 1 Πs * λ when t = 1 in terms of what we call γ-parking functions and lattice γ-parking functions. Here, ∆F and Π are certain eigenoperators of the modified Macdonald basis and M = (1 − q)(1 − t). Our main results in turn give an elementary basis expansion at t = 1 for symmetric functions of the form M ∆F e 1 ΘGJ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis {Πe * λ } λ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

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A proof of the fermionic Theta coinvariant conjecture

Cited by 3 publications

References 18 publications

Quasisymmetric harmonics of the exterior algebra

Quasisymmetric harmonics of the exterior algebra

Quasisymmetric harmonics of the exterior algebra

Delta and Theta Operator Expansions

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