[SAL] From the extended Schur algebra to the rook monoid | Inês Legatheaux Martins (CMAFcIO, FCUL)
20 May 2024 2:00 pm – 3:00 pm
DM Meeting Room, Building VII
Abstract:
Schur algebras were defined in 1980 by J. A. Green in a seminal monograph. Given an infinite field $\mathbb{F}$, they provide a natural setting for understanding the (polynomial) representations of the general linear group $GL_d(\mathbb{F})$ and their interactions with those of the symmetric group $S_n$. In particular, they were pivotal for extending the classical Schur-Weyl duality for $GL_d(\mathbb{F})$ and $S_n$ on tensors to positive characteristic.
Recently, we introduced the extended Schur algebra $\mathcal{S}_{\mathbb{F}} (d,\mathbf{n})$ and proved that its module category is equivalent to the category of finite-dimensional polynomial $GL_d(\mathbb{F})$-modules which are homogeneous of degree at most $n$. Provided that $d \geq n$, and $\mathbb{F}$ has characteristic zero, we gave a new instance of Schur-Weyl duality for $\mathcal{S}_{\mathbb{F}} (d,\mathbf{n})$ and an important generalisation of $S_n$ called the rook monoid $R_n$ (also known as the symmetric inverse monoid).
Another upshot of Green’s monograph is the description of several functors relating the module categories of the algebras $A$ and $eAe$, where $e \in A$ is an idempotent. The aim of this talk is to give some applications of this general theory to the modular representation theory of the rook monoid.
Our purpose is to explain how these methods can be used to show that the Schur-Weyl duality between $\mathcal{S}_{\mathbb{F}} (d,\mathbf{n})$ and $R_n$ remains true for an infinite field $\mathbb{F}$ of arbitrary characteristic as well as to give a combinatorial construction of certain $\mathbb{F}R_n$-modules which are analogous to the dual Specht modules for $S_n$. In good characteristic, this construction will form a complete set of simple modules for the rook monoid.
[SSRM, SOR] Developing Optimization models in urban transportation fleet | Parisa Ahani (NOVA Math)
22 May 2024 2:00 pm – 3:00 pm
Title: Developing Optimization models in urban transportation fleet
Speaker: Parisa Ahani, NOVA Math
Date | Time: May 22, 2024 | 14h00
Place: FCT NOVA, VII-Second Floor, Seminar room
Abstract: Recently, the use of more sustainable forms of transportation such as electric vehicles (EVs) for delivering goods and parcels to customers in urban areas has received more attention from urban planners and private stakeholders. The urban freight transportation sector is examining such a shift toward using electric vehicles, besides current combustion engine vehicles, to deliver goods and services to customers. To contribute toward sustainable transportation in urban logistics, we consider the important factor of decision replacement management and study how to shift toward sustainable modes of transportation, specifically EVs, in an urban area. We will present various optimization frameworks for different vehicle replacement decision problems that can be used by the operators (private and public stakeholders) for a combination of various types of vehicles in their fleet in order to achieve an optimal fleet structure.
Organizers: Isabel Natário & Mina Norouzirad & Graça Gonçalves
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This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications)
NOVA Math‘s focus is on cutting edge research, in both pure and applied mathematics, valuing the use of mathematics in the solution of real-world problems at the industrial level and of social relevance.
One of the main strategies developed by NOVA Math is to promote the exchange of knowledge with other sciences. It is important to engage with the users of mathematics, given them the support for their research on one hand, and on another hand, to direct mathematical researchers that seek real-life problems.
Funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the following projects:
UIDB/00297/2020, UIDP/00297/2020, UID/MAT/00297/2019, UID/MAT/00297/2013, PEst-OE/MAT/UI0297/2014, PEst-OE/MAT/UI0297/2011.