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A level-two cyclotomic polynomial from the highest-weight filtration

Example

Assume the inherited Axiom of Choice. Take source-ordered row lengths $p = (1, 2)$ , so the column heights are $q = (2, 1)$ , and let $N = 3$ , $n = 2$ , $c = (0, 0)$ . Number the boxes down the columns: $1 = (1, 1)$ , $2 = (2, 1)$ , $3 = (2, 2)$ . The pyramid nilpotent is $e = E_{23}$ , and on the one-step deformed tensor space $V_{c}^{\otimes 1} = V = C^{3}$ the operator $x_{1}$ is $x_{1} = E_{23} + diag (0, 0, - 1) .$ Its eigenvalue $0$ has multiplicity $2$ and eigenvalue $- 1$ has multiplicity $1$ , with $- 1$ -eigenspace $C (v_{3} - v_{2})$ ; its minimal polynomial is $x (x + 1)$ , with roots $a = (0, - 1)$ , and $Λ = Λ_{0} + Λ_{- 1}$ . For every $d \geq 1$ the raw tensor operator $x_{1}$ on $V_{c}^{\otimes d}$ still has exactly this minimal polynomial, by the column-filtration and leading-string argument. Consequently $H_{d} (Λ)$ has dimension $2^{d} d!$ , and at $d = 1$ its image splits the two stated eigenspaces.

Facts & Assumptions

Given: The pyramid with columns $q = (2, 1)$ , $n = 2$ , $N = 3$ , origin $c = (0, 0)$ , the numbers $a = (0, - 1)$ , and the operator $x_{1}$ on $V_{c}^{\otimes d}$ .

[F1]

On the raw tensor space the operator $x_{1}$ is $(e + \sum_{j} (c_{c o l (j)} + q_{c o l (j)} - n) e_{j, j}) \otimes 1^{\otimes (d - 1)}$ minus its off-diagonal cross-column corrections; its minimal polynomial on $V_{c}^{\otimes d}$ is exactly $\prod_{j} (x - a_{j})$ with $a_{j} = c_{j} + q_{j} - n$ , and the tensor action of $H_{d}$ factors through $H_{d} (a)$ (The parabolic highest-weight truncation factors through the cyclotomic quotient, Arakawa-Suzuki Casimir operators on tensor functors).

[F2]

The cyclotomic quotient $H_{d} (a)$ of the degenerate affine Hecke algebra at the roots $a = (0, - 1)$ has the bounded monomial basis $x_{1}^{r_{1}} \dots x_{d}^{r_{d}} w$ , $0 \leq r_{i} < 2$ , $w \in S_{d}$ , so that $\dim H_{d} (a) = 2^{d} d!$ ; the pyramid roots are $a_{j} = c_{j} + q_{j} - n$ , and the integral-root dictionary gives $Λ = Λ_{0} + Λ_{- 1}$ (The degenerate cyclotomic Hecke algebra).

Proof

technique · direct

Given: The boxes $1 = (1, 1)$ , $2 = (2, 1)$ , $3 = (2, 2)$ , the nilpotent $e = E_{23}$ , and the roots $a_{1} = 0 + 2 - 2 = 0$ , $a_{2} = 0 + 1 - 2 = - 1$ .

1.1F1algebra

The operators on $V$ . For $d = 1$ the cross-column correction in [F1] is absent and $\sum_{j} (c_{c o l (j)} + q_{c o l (j)} - n) e_{j, j} = diag (0, 0, - 1)$ , so $x_{1} = E_{23} + diag (0, 0, - 1)$ : explicitly $x_{1} v_{1} = 0$ , $x_{1} v_{2} = 0$ and $x_{1} v_{3} = v_{2} - v_{3}$ . Hence $x_{1} (x_{1} + 1) = 0$ , the eigenspaces of $x_{1}$ are $C v_{1} \oplus C v_{2}$ for the eigenvalue $0$ and $C (v_{3} - v_{2})$ for the eigenvalue $- 1$ , and the minimal polynomial is exactly $x (x + 1)$ : $x_{1} \neq 0$ since $x_{1} v_{3} = v_{2} - v_{3} \neq 0$ , and $x_{1} + 1 \neq 0$ since $(x_{1} + 1) v_{1} = v_{1} \neq 0$ .

2.1F1F2step 1.1algebra∎

All degrees and the cyclotomic quotient. By the minimal-polynomial clause of [F1], the operator $x_{1}$ on $V_{c}^{\otimes d}$ has minimal polynomial $\prod_{j} (x - a_{j}) = x (x + 1)$ for every $d \geq 1$ , so the tensor action factors through $H_{d} (a)$ with $a = (0, - 1)$ , i.e. through $H_{d} (Λ)$ for $Λ = Λ_{0} + Λ_{- 1}$ ; and by [F2] that quotient has dimension $2^{d} d!$ . At $d = 1$ the quotient is $C [x] / (x (x + 1)) ≅ C \times C$ , acting on $V$ through $x_{1}$ as the two-dimensional algebra of operators that are scalar on each of the two eigenspaces of step 1.1; hence the image is semisimple and $V$ is the direct sum of those two eigenspaces.

Remarks

The two roots $0$ and $- 1$ are exactly the column scalars $c_{j} + q_{j} - n$ of the two columns; the example shows how the highest-weight (column) filtration of the ground module produces the level-two cyclotomic polynomial $x (x + 1)$ .
The ordered-row dictionary matters here: the row lengths are read in the source order $p = (1, 2)$ with column heights $q = (2, 1)$ , so box $3$ is the rightmost box of the second row and $e = E_{23}$ ; a different box numbering would produce a different matrix for $x_{1}$ .

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Sources

Jonathan Brundan and Alexander Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. (N.S.) 14 (2008) 1-57 (complete 50-page arXiv article) (standard reference, not scraped)
Jonathan Brundan and Alexander Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010) 877-919 (complete 44-page arXiv article) (standard reference, not scraped)