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Steen Ryom-Hansen

Publications and source records attributed to Steen Ryom-Hansen.

At least 19 recordsLinked to original sources
arXiv2025

A KLR-like presentation for the bt-algebra

We consider the bt-algebra ${ \mathcal E}_n(q)$ of knot theory, defined over an arbitrary field $ \Bbbk$. We find a KLR-like presentation for $ {\mathcal E}_n(q) $ showing that it is a $ \mathbb Z$-graded algebra if $ q \in \Bbbk^{\times} \setminus \{1 \} $ admits a square root in $ \Bbbk $. We introduce the ordered bt-algebra $ {\mathcal E}^{\rm{ord}}_n(q)$ and show that it also has a KLR-like presentation, without restriction on $ q \in \Bbbk^{\times} \setminus \{1 \} $. In particular, $ {\mathcal E}^{\rm{ord}}_n(q)$ is a $ \mathbb Z$-graded algebra for all $ q \in \Bbbk^{\times} \setminus \{1 \} $.

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arXiv2024

On the spherical partition algebra

For $ k \in \mathbb{N}$ we introduce an idempotent subalgebra, the spherical partition algebra ${\mathcal{SP} }_{k}$, of the partition algebra ${\mathcal{P} }_{k}$, that we define using an embedding associated with the trivial representation of the symmetric group $\mathfrak{S}_k$. We determine a basis for ${\mathcal{SP} }_{k}$ and this provides a combinatorial interpretation of the dimension of $\mathcal{SP}_{k}$, involving bipartite partitions of $ k$. For $ t \in \mathbb{C} $ we consider the specialized algebra $\mathcal{SP}_{k}(t)$. For $ t = n \in \mathbb{N}$, we describe the structure of $\mathcal{SP}_{k}(n)$ by giving the permutation module decomposition of the $k$'th symmetric power of the defining module for the symmetric group algebra $ \mathbb{C} \mathfrak{S}_n $. In general, we show that $\mathcal{SP}_{k}(t)$ is quasi-hereditary over $ \mathbb{C}$ for all $ t \in \mathcal{C}$, except $ t=0$. We determine the decomposition numbers for $\mathcal{SP}_{k}(t)$ for every specialization $ t \in \mathbb{C} $ except $ t= 0 $, (which includes semisimple and non-semisimple cases). In particular we determine the structure of all indecomposable projective modules, and the indecomposable tilting modules.

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arXiv2024

Seminormal forms for the Temperley-Lieb algebra

Let ${\mathbb{TL}_n^{\! \mathbb Q}} $ be the rational Temperley-Lieb algebra, with loop parameter $ 2 $. In the first part of the paper we study the seminormal idempotents $ E_{ \mathfrak{t}} $ for ${\mathbb{TL}_n^{\! \mathbb Q}}$ for $ \mathfrak{t} $ running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of $ E_{\mathfrak{t}} $ using Jones-Wenzl idempotents $ {\mathbf{JW}_{\! k}} $ for ${\mathbb{TL}_k^{\! \mathbb Q}}$ where $ k \le n $. In the second part of the paper we consider the Temperley-Lieb algebra ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ over the finite field $ {\mathbb F}_p$, where $ p>2$. The KLR-approach to ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$ gives rise to an action of a symmetric group $ \mathfrak{S}_m$ on ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$, for some $ m < n $. We show that the $ E_{ \mathfrak{t}} $'s from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for $ \mathfrak{S}_m$. This leads to a KLR-interpretation of the $p$-Jones-Wenzl idempotent $ ^{p}\!{\mathbf{JW}_{\! n}} $ for ${\mathbb{TL}_n^{\! {\mathbb F}_p}}$, that was introduced recently by Burull, Libedinsky and Sentinelli.

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arXiv2023

Graded sum formula for $\tilde{A}_1$-Soergel calculus and the nil-blob algebra

We study the representation theory of the Soergel calculus algebra $ A_w := \mbox{End}_{{\mathcal D}_{(W,S)}} (\underline{w}) $ over $\mathbb C$ in type $\tilde{A}_1$. We generalize the recent isomorphism between the nil-blob algebra ${\mathbb{NB}}_n$ and $ A_w $ to deal with the two-parameter blob algebra. Under this generalization, the two parameters correspond to the two simple roots for $\tilde{A}_1$. Using this, together with calculations involving the Jones-Wenzl idempotents for the Temperley-Lieb subalgebra of $ \mathbb{NB}_n$, we obtain a concrete diagonalization of the matrix of the bilinear form on the cell module $Δ_w(v) $ for $ A_w $. The entries of the diagonalized matrices turn out to be products of roots for $\tilde{A}_1$. We use this to study Jantzen type filtrations of $ Δ_w(v)$ for $A_w $. We show that at enriched Grothendieck group level the corresponding sum formula has terms $ Δ_w(s_{α}v)[ l(s_{α}v)- l(v)] $, where $[ \cdot ] $ denotes grading shift.

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arXiv2022

On the denominators of Young's seminormal basis

We study the seminormal basis ${f_t}$ for the Specht modules of the Iwahori-Hecke algebra ${\cal H}_n(q)$ of type $A_{n-1}$. We focus on the base change coefficients between the seminormal basis ${f_t}$ and Young's natural basis ${x_t}$ with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving radial lengths. Even for general tableaux we obtain new formulas. On the way we prove a new result about summands of the restricted Specht module at root of unity.

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arXiv2021

On the annihilator ideal in the $bt$-algebra of tensor space

We study the representation theory of the braids and ties algebra, or the $bt$-algebra, $ \cal E$. Using the cellular basis $\{m_{{\mathfrak s} {\mathfrak t}} \}$ for $ \cal E$ obtained in previous joint work with J. Espinoza we introduce two kinds of permutation modules $M(λ)$ and $ M(Λ) $ for $\cal E$. We show that the tensor product module $V^{\otimes n}$ for $\cal E$ is a direct sum of $ M(λ)$'s. We introduce the dual cellular basis $\{n_{{\mathfrak s} {\mathfrak t}} \}$ for $ \cal E $ and study its action on $ M(λ) $ and $ M(Λ) $. We show that the annihilator ideal $ \cal I $ in $ \cal E $ of $ V^{\otimes n } $ enjoys a nice compatibility property with respect to $\{n_{{\mathfrak s} {\mathfrak t}} \}$. We finally study the quotient algebra $ {\cal E}/{\cal I} $, showing in particular that it is a simultaneous generalization of Härterich's 'generalized Temperley-Lieb algebra' and Juyumaya's 'partition Temperley-Lieb algebra'.

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arXiv2020

The nil-blob algebra: An incarnation of type $\tilde{A}_1$ Soergel calculus and of the truncated blob algebra

We introduce a type $B$ analogue of the nil Temperley-Lieb algebra in terms of generators and relations, that we call the (extended) nil-blob algebra. We show that this algebra is isomorphic to the endomorphism algebra of a Bott-Samelson bimodule in type $\tilde{A}_1$. We also prove that it is isomorphic to an idempotent truncation of the classical blob algebra.Thus we provide strong evidence in favor of the recent Blob vs. Soergel conjecture.

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arXiv2019

Graded cellular basis and Jucys-Murphy elements for generalized blob algebras

We give a concrete construction of a graded cellular basis for the generalized blob algebra B_n introduced by Martin and Woodcock. The construction uses the isomorphism between KLR-algebras and cyclotomic Hecke algebras, proved by Brundan-Kleshchev and Rouquier. It gives rise to a family of Jucys-Murphy elements for B_n.

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arXiv2015

Projective modules for the symmetric group and Young's seminormal form

We study the representation theory of the symmetric group $S_n$ in positive characteristic $p$. Using features of the LLT-algorithm we give a conjectural description of the projective cover $P(λ)$ of the simple module $D(λ)$ where $λ$ is a $p$-restricted partition such that all ladders of the corresponding ladder partition are of order less than $p$. Inspired by the recent theory of Khovanov-Lauda-Rouquier algebras we explain an algorithm that allows us to verify this conjectural description for $n \leq 15$, at least.

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arXiv2013

Graded cellular bases for Temperley-Lieb algebras of type A and B

We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.

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arXiv2012

Cell structures on the blob algebra

We consider the $ r = 0 $ case of the conjectures by Bonnafé, Geck, Iancu and Lam on cellular structures on the Hecke algebra of type $ B $. We show that this case induces the natural cell structure on the blob algebra $ b_n $ by restriction to one-line bipartitions.

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arXiv2012

Young's seminormal form and simple modules for $S_n$ in characteristic $p$

We realize the integral Specht modules for the symmetric group $S_n$ as induced modules from the subalgebra of the group algebra generated by the Jucys-Murphy elements. We deduce from this that the simple modules for $FS_n$ are generated by reductions modulo $p$ of the corresponding Jucys-Murphy idempotents.

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arXiv2010

On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.

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arXiv2010

The Ariki-Terasoma-Yamada tensor space and the blob-algebra

We show that the Ariki-Terasoma-Yamada tensor module and its permutation submodules $ M(λ) $ are modules for the blob algebra when the Ariki-Koike algebra is a Hecke algebra of type $B$. We show that $ M(λ)$ and the standard modules $ Δ(λ) $ have the same dimensions, the same localization and similar restriction properties and are equal in the Grothendieck group. Still we find that the universal property for $ Δ(λ) $ fails for $ M(λ) $, making $ M(λ) $ and $ Δ(λ) $ different modules in general. Finally, we prove that $ M(λ) $ is isomorphic to the dual Specht module for the Ariki-Koike algebra.

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arXiv2009

Some remarks on Ext groups

We calculate certain ext-groups between modules for a linear algebraic group. The results are in agreement with the Lusztig conjecture.

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arXiv2009

Koszul duality of translation--and Zuckerman functors

We review Koszul duality in representation theory of category $ \cal O $, especially we give a new presentation of the Koszul duality functor. Combining this with work of Backelin, we show that the translation and Zuckerman functors are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel and Khovanov. Finally we use Koszul duality to give a short proof of the Enright-Shelton equivalence.

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arXiv2009

The Lascoux, Leclerc and Thibon algorithm and Soergel's tilting algorithm

We generalize Soergel's tilting algorithm to singular weights and deduce from this the validity of the Lascoux-Leclerc-Thibon conjecture on the connection between the canonical basis of the basic submodule of the Fock module and the representation theory of the Hecke-algebras at root of unity.

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