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Semisimplicity criterion for 2-tonal partition algebras

C. Ahmed, G. M. Benkart, O. H. King, P. P. Martin, A. E. Parker

Abstract

We determine the semisimplicity criterion for even partition algebras over the complex field. Specifically we prove that the even/2-tonal partition algebras $P_n^2(δ)$ over $\mathbb{C}$ are semisimple for all $n$ if and only if parameter $δ\not\in \mathbb{N}_0$ .

Semisimplicity criterion for 2-tonal partition algebras

Abstract

We determine the semisimplicity criterion for even partition algebras over the complex field. Specifically we prove that the even/2-tonal partition algebras over are semisimple for all if and only if parameter .

Paper Structure

This paper contains 12 sections, 7 theorems, 71 equations, 4 figures.

Table of Contents

  1. Outline
  2. Definitions and basic properties of tonal partition algebras $P_n^d(\delta)$
  3. The tonal partition algebras $P_n^d(\delta)$
  4. Basic representation theory
  5. Gram matrix generalities
  6. Potts functors, symmetry and simple modules
  7. Proof of semisimplicity condition for the ordinary $d =1$ case recalled
  8. Gram matrices, tower representation theory and semisimplicity
  9. General preparations for the 2-tonal ($d=2$) case
  10. Potts functor preparations for the tonal cases
  11. The 2-tonal ($d=2$) case
  12. Gram matrices for $P^2_n$ and the main Theorem

Key Result

Lemma 4.6

Let $\delta =l \in\mathbb{N}$, and $n$ even. Then (the sum runs from $t=0$ to include the $n=0$ case). ∎ $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Augmented Bratelli diagram for algebras $P_0 \subset P_1 \subset ...\subset P_4$. Vertices are standard modules with index as shown at the top of their column; and dimension shown in the box. Black edges indicate restriction rules, with multiplicities, so dimensions can be checked. Green arrows indicate module morphisms for $\delta=1 \in \mathbb{C}$ (see main text for commentary). Pink arrows indicate morphisms for $\delta=2$. (Morphisms for other $\delta$s omitted to avoid clutter.)
  • Figure 2: Augmented Bratelli diagram for $P^2_n(\delta)$ up to $n=4$ (cf. Fig.\ref{['fig:PnBratelli']}), but here truncated to exclude some modules at rank 4, such as for $\lambda = \left( (1^2),(1)\right)$. The coloured arrows represent the maps between the standard modules for the specified $\delta$. For example, on level $n=4$ when $\delta=1$ there is a map from $\mathcal{S}_4(\emptyset, (2))$ to $\mathcal{S}_{4}(\emptyset,\emptyset)$, indicated by a green arrow, with image of dimension 3. $\;$ (See (\ref{['pa:Trep']}).)
  • Figure 3: Gram matrices for $\mathcal{S}_n(0)$ for $n=2,3,4$. The bases are $\{ (1)(2), (12) \}$, $\{ (1)(2)(3), (12)(3), (13)(2), (1)(23), (123) \}$, $\{ (1)(2)(3)(4), (12)(3)(4), (13)(2)(4),$$(1)(23)(4), (14)(2)(3), (1)(24)(3), (1)(2)(34), (123)(4), (124)(3), (12)(34), ..., (1234) \}$ respectively (see main text for notation).
  • Figure 4: The first matrix is the Gram matrix of $\langle-,- \rangle_{e_{(\emptyset,\emptyset)}}$ when $n=4$, with respect to the indicated basis; $\;$ and the second is the gram matrix of $\langle-,- \rangle_{e_{((1),\emptyset)}}$ when $n=3$.

Theorems & Definitions (15)

  • Example 2.31
  • Lemma 4.6
  • Lemma 5.6
  • proof
  • Lemma 5.8
  • proof
  • Lemma 5.12
  • proof
  • Lemma 5.13
  • proof
  • ...and 5 more