A conjecture of a basis for the diagonal harmonic alternants

Allen, Edward E.

doi:10.1016/s0012-365x(98)90013-9

Cited by 1 publication

(11 citation statements)

References 4 publications

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“…This identity reveals that the alternating character χ [1,1,1] occurs in bi-degrees (3, 0), (2, 1), (1, 2), (0, 3) and (1,1). In this particular case these are Frobenius images of the sl [2] strings generated by the following two alternants…”

Section: I13mentioning

confidence: 89%

“…Let us call it String 1. String 2 starts with bi-degree (8, 1) and ends in bi-degree (1,8). String 3 starts with bi-degree (7, 1) and ends in bi-degree (1,7).…”

Section: Proof Of Theorem I1mentioning

confidence: 99%

“…String 2 starts with bi-degree (8, 1) and ends in bi-degree (1,8). String 3 starts with bi-degree (7, 1) and ends in bi-degree (1,7). String 4 starts with bi-degree (6, 2) and ends in bi-degree (2,6).…”

Section: Proof Of Theorem I1mentioning

confidence: 99%

“…That accounts for the 2 ′ s in the image of its path. String 5 starts with bi-degree (6, 1) and ends in bi-degree (1,6). These two ends are ∆ 21 3 and ∆ 41 .…”

Section: Proof Of Theorem I1mentioning

confidence: 99%

“…Conjecture in [1] and Theorem 2.1, give us an algorithm for constructing starters, this done a basis for DHA n is easily constructed. From the display in 2.20, we see that the number of starters for n = 7 is 44 and for n = 8 is already 120.…”

Section: 20mentioning

confidence: 99%

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A basis for the Diagonal Harmonic Alternants

Garsia¹,

Zabrocki²

2022

Preprint

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It will be shown here that there are differential operators E, F and H = [E, F ] for each n ≥ 1, acting on Diagonal Harmonics, yielding that DH n is a representation of sl[2] (see [3] Chapter 3). Our main effort here is to use sl[2] theory to predict a basis for the Diagonal Harmonic Alternants, DHA n . It can be shown that the irreducible representations sl[2] are all of the form P, EP, E 2 P, • • • , E k P , with F P = 0 and E k+1 P = 0. The polynomial P is known to be called a "String Starter". From sl[2] theory it follows that DHA n is a direct sum of strings. Our main result so far is a formula for the number of string starters. A recent paper by Carlsson and Oblomkov (see [2]) constructs a basis for the space of Diagonal Coinvariants by Algebraic Geometrical tools. It would be interesting to see if any our results can be derived from theirs.

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Section: I13mentioning

confidence: 89%

“…Let us call it String 1. String 2 starts with bi-degree (8, 1) and ends in bi-degree (1,8). String 3 starts with bi-degree (7, 1) and ends in bi-degree (1,7).…”

Section: Proof Of Theorem I1mentioning

confidence: 99%

Section: Proof Of Theorem I1mentioning

confidence: 99%

“…That accounts for the 2 ′ s in the image of its path. String 5 starts with bi-degree (6, 1) and ends in bi-degree (1,6). These two ends are ∆ 21 3 and ∆ 41 .…”

Section: Proof Of Theorem I1mentioning

confidence: 99%

Section: 20mentioning

confidence: 99%

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