More Like this

1. GENERICALLY FREE REPRESENTATIONS II: IRREDUCIBLE REPRESENTATIONS

   https://doi.org/10.1007/s00031-020-09591-3  

   Garibaldi, S.; Guralnick, R. (January 2020, Transformation groups)

   We determine which faithful irreducible representations V of a simple linear algebraic group G are generically free for Lie(G), i.e., which V have an open subset consisting of vectors whose stabilizer in Lie(G) is zero. This relies on bounds on dim V obtained in prior work (part I), which reduce the problem to a finite number of possibilities for G and highest weights for V , but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position. 

   more » « less
2. Generically free representations I: Large representations

   https://doi.org/doi.org/10.2140/ant.2020.14.1577  

   Garibaldi, S.; Guralnick, R. (January 2020, Algebra and number theory)

   This paper concerns a faithful representation V of a simple linear algebraic group G. Under mild assumptions, we show that if V is large enough, then the Lie algebra of G acts generically freely on V. That is, the stabilizer in Lie.G/ of a generic vector in V is zero. The bound on dim V grows like the square of the rank and holds with only mild hypotheses on the characteristic of the underlying field. The proof relies on results on generation of Lie algebras by conjugates of an element that may be of independent interest. We use the bound in subsequent works to determine which irreducible faithful representations are generically free, with no hypothesis on the characteristic of the field. This in turn has applications to the question of which representations have a stabilizer in general position. 

   more » « less
3. Temperature dependent charge transport in electrostatically doped poly[benzimidazobenzophenanthroline] thin films

   https://doi.org/10.1002/app.53470  

   Cruz‐Arzon, Alejandro J.; Serrano‐Garcia, William; Pinto, Nicholas J.; Gupta, Nikita; Johnson, Alan T. Charlie (November 2022, Journal of Applied Polymer Science)

   Abstract Charge transport in electrostatically doped poly[benzimidazobenzophenanthroline]‐BBL thin films in a field‐effect transistor geometry were investigated in the temperature range 150 K < T < 370 K. At low temperatures activation and hopping transport mechanisms dominated, while phonon scattering dominated at high temperatures. The activation energies (E_A) were found to lie in the range 140 meV < E_A < 400 meV implying the existence of deep traps within the polymer bandgap of 1.8 eV. Two quasi‐linear dependencies ofE_Aon the gate voltage (V_g) were observed withE_Adecreasing asV_gincreased. An unexpected “metallic‐like” transport characteristic appeared forT > 335 K which depended onV_g. Enhanced electron delocalization combined with increased carrier density could be responsible for this “metallic‐like” behavior. Our results show that the existence of deep traps with multiple energy distributions, combined with increased carrier density led to the unusual temperature dependence of charge transport observed in BBL. 

   more » « less
4. Graphs with no even holes and no sector wheels are the union of two chordal graphs

   https://doi.org/10.1016/j.ejc.2024.104035  

   Abrishami, Tara; Berger, Eli; Chudnovsky, Maria; Zerbib, Shira (December 2024, European Journal of Combinatorics)

   Sivaraman (2020) conjectured that if G is a graph with no induced even cycle then there exist sets X1,X2⊆V(G) satisfying V(G)=X1∪X2 such that the induced graphs G[X1] and G[X2] are both chordal. We prove this conjecture in the special case where G contains no sector wheel, namely, a pair (H,w) where H is an induced cycle of G and w is a vertex in V(G)∖V(H) such that N(w)∩H is either V(H) or a path with at least three vertices. 

   more » « less
5. Universal Tensor Categories Generated by Dual Pairs

   https://doi.org/10.1007/s10485-021-09640-2  

   Chirvasitu, Alexandru; Penkov, Ivan (October 2021, Applied Categorical Structures)

   Abstract Let $$V_*\otimes V\rightarrow {\mathbb {C}}$$ V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V ∗ . The Mackey Lie algebra $${\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ∗ ) corresponding to this pairing consists of all endomorphisms $$\varphi $$ φ of V for which the space $$V_*$$ V ∗ is stable under the dual endomorphism $$\varphi ^*: V^*\rightarrow V^*$$ φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category $${\mathbb {T}}$$ T generated by the $${\mathfrak {g}}$$ g -modules V , $$V_*$$ V ∗ and their algebraic duals $$V^*$$ V ∗ and $$V^*_*$$ V ∗ ∗ . The category $${{\mathbb {T}}}$$ T is an analogue of categories considered in prior literature, the main difference being that the trivial module $${\mathbb {C}}$$ C is no longer injective in $${\mathbb {T}}$$ T . We describe the injective hull I of $${\mathbb {C}}$$ C in $${\mathbb {T}}$$ T , and show that the category $${\mathbb {T}}$$ T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category $$_I{\mathbb {T}}$$ I T of objects in $${\mathbb {T}}$$ T which are free as I -modules. Our main result is that the category $${}_I{\mathbb {T}}$$ I T is also Koszul, and moreover that $${}_I{\mathbb {T}}$$ I T is universal among abelian $${\mathbb {C}}$$ C -linear tensor categories generated by two objects X , Y with fixed subobjects $$X'\hookrightarrow X$$ X ′ ↪ X , $$Y'\hookrightarrow Y$$ Y ′ ↪ Y and a pairing $$X\otimes Y\rightarrow {\mathbf{1 }}$$ X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $${\mathbb {T}}$$ T and $${}_I{\mathbb {T}}$$ I T . 

   more » « less