Subalgebras of Lie algebra of Lorentz group will be discussed as the basic examples at the beginning of the report. The corresponding analysis is performed by canonical bases for subspaces of a vector space. All canonical bases for 5-dimensional and 4-dimensional subspaces of a 6-dimensional vector space are found, and they are utilized to find…
Discovered in the 1730s, the classical Euler-Maclaurin formula may be viewed as a formula for summing the values of a function over the lattice points in a one-dimensional polytope. Several years ago, Berline and Vergne generalized this formula to polytopes of arbitrary dimension, obtaining a formula for the sum of a polynomial function over the…
The Schur-Weyl duality between the symmetric group and the general linear group allows us to connect the representation theory of these two groups. A consequence of this duality is the Frobenius formula which connects the irreducible characters of the general linear group and the symmetric group via symmetric functions. The symmetric group is also in…
Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a…