Department of Mathematics, Kyoto Prefectural University of Medicin

Ikumitsu Nagasaki

長崎 生光

English

長崎, 生光

ナガサキ, イクミツ

Nagasaki, Ikumitsu

Kyoto Prefectural University of Medicine Institutional Repository (Kissei)

An estimate of the isovariant Borsuk-Ulam constant for a group of type B2Ikumitsu NAGASAKI 1)Abstract. The isovariant Borsuk-Ulam constant cG of a compact Lie group Gis defined to be the supremum of constants c ∈ R with the following property: Ifthere exists a G-isovariant map f : S(V ) → S(W ), thenc(dimV − dimV G) ≤ dimW − dimWGholds. Several estimates of cG are known. In our previous study, we provided anew estimate of cG for G of type A2. In this paper, we treat a group of type B2and provide a new estimate of cG by using representation theory.1. IntroductionLet G be a compact Lie group, and S(V ) and S(W ) the unit spheres of (orthogo-nal) G-representations V and W respectively. A G-map f : S(V ) → S(W ) is calledisovariant if it preserves the isotropy groups.Definition. The isovariant Borsuk-Ulam constant cG of a compact Lie group G isdefined to be the supremum of constants c ∈ R with the following property: If thereexists a G-isovariant map f : S(V ) → S(W ), thenc(dimV − dimV G) ≤ dimW − dimWGholds.The determination of cG is an interesting and important problem for the study of theisovariant Borsuk-Ulam type theorem; however, it is difficult at present and thereforewe shall provide estimates of cG. For any connected compact Lie group, we havealready provided some mild estimates in [3]; for example, 34 ≤ cG ≤ 1 for G of type A2and 45 ≤ cG ≤ 1 for G of type B2. In [4], we provided a better estimate 2627 ≤ cG ≤ 1for type A2. The goal of this paper is to provide a better estimate for type B2; namely,we shall show1) Department of Mathematics, Kyoto Prefectural University of Medicine, 1-5 Shimogamohangi-cho, Sakyo-ku, Kyoto 606-0823, Japan. e-mail: nagasaki@koto.kpu-m.ac.jp2010 Mathematics Subject Classification. Primary 57S15; Secondary 57S25, 55M20, 22E99.An estimate of the isovariant Borsuk-Ulam constant for a group of type B 2 41Theorem 1.1. Let G be of type B2. Then1314 ≤ cG ≤ 1.2. From representation theoryWe collect necessary facts from representation theory in order to obtain the newestimate. Since representation theory of connected compact Lie groups is equivalentto that of complex Lie groups or algebraic groups, we may considerG = B2 = {A ∈ SL5(C) | tAJA = J} ∼= SO5(C)as a group of type B2, whereJ =⎛⎜⎜⎜⎝0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0⎞⎟⎟⎟⎠.Note that a maximal compact subgroup of G is isomorphic to SO(5) and a maximaltorus T of G can be taken as the group consisting of all elementst =⎛⎜⎜⎜⎝t1 0t21t−120 t−11⎞⎟⎟⎟⎠for t1, t2 ∈ C∗, which is isomorphic to C∗ × C∗.Irreducible representations of G are parametrized by the dominant weights. In typeB2, the set of dominant weights is given byΛ+ = {λ = (λ1, λ2) ∈ Z2 |λ1 ≥ λ2 ≥ 0}.We denote by V (λ) the irreducible representation corresponding to λ ∈ Λ+. Considerelements of G:a =⎛⎜⎜⎜⎝0 0 0 0 10 1 0 0 00 0 −1 0 00 0 0 1 01 0 0 0 0⎞⎟⎟⎟⎠, b =⎛⎜⎜⎜⎝1 0 0 0 00 0 0 1 00 0 −1 0 00 1 0 0 00 0 0 0 1⎞⎟⎟⎟⎠, c =⎛⎜⎜⎜⎝0 1 0 0 01 0 0 0 00 0 1 0 00 0 0 0 10 0 0 1 0⎞⎟⎟⎟⎠belonging to N = NG(T ). The subgroup W generated by a, b, c is isomorphic to thedihedral group D4 of order 8 and N is isomorphic to T W . Therefore the Weyl groupNG(T )/T of G is isomorphic to W . Thus we identify W with the Weyl group of G.Ikumitsu NAGASAKI42The conjugate action of W on T is given bya⎛⎜⎜⎜⎝t1 0t21t−120 t−11⎞⎟⎟⎟⎠a−1 =⎛⎜⎜⎜⎝t−11 0t21t−120 t1⎞⎟⎟⎟⎠,b⎛⎜⎜⎜⎝t1 0t21t−120 t−11⎞⎟⎟⎟⎠b−1 =⎛⎜⎜⎜⎝t1 0t−121t20 t−11⎞⎟⎟⎟⎠,c⎛⎜⎜⎜⎝t1 0t21t−120 t−11⎞⎟⎟⎟⎠c−1 =⎛⎜⎜⎜⎝t2 0t11t−110 t−12⎞⎟⎟⎟⎠.There are five conjugacy classes of elements of W given by(1), (a) = (b), (c) = (abc), (ab), (ac) = (bc).The subgroups and the conjugacy classes of subgroups of W are described in figures 1and 2.W〈a, b〉~~~~ 〈ac〉 〈ab, c〉 〈a〉 〈b〉〈ab〉 〈c〉〈abc〉																		1Figure 1. The subgroup lattice of WThere are five irreducible representations of W , say C, Csgn, C−, Csgn− and U ,whose characters are described in table 1.An estimate of the isovariant Borsuk-Ulam constant for a group of type B 2 43W(〈a, b〉)(〈ac〉) (〈ab, c〉)(〈a〉)(〈ab〉) (〈c〉)(1)Figure 2. The lattice of conjugacy classes(1) (a) (ab) (ac) (c)C 1 1 1 1 1Csgn 1 −1 1 −1 1C− 1 1 1 −1 −1Csgn− 1 −1 1 1 −1U 2 0 −2 0 0Table 1. The character table of WFrom the character table, we see the following.Proposition 2.1. There are the following isomorphisms as W -representations.(I) (a) C[W/W ] ∼= C.(b) C[W/〈a〉] ∼= C[W/〈b〉] ∼= C⊕ C− ⊕ U .(c) C[W/〈a, b〉] ∼= C⊕ C−.(d) C[W/〈c〉] ∼= C[W/〈abc〉] ∼= C⊕ Csgn ⊕ C− ⊕ U .(e) C[W ] ∼= C⊕ Csgn ⊕ C− ⊕ Csgn− ⊕ 2U .(II) (a) C[W/W ]⊗ Csgn ∼= Csgn.(b) C[W/〈a〉]⊗ Csgn ∼= Csgn ⊕ Csgn− ⊕ U .(c) C[W/〈a, b〉]⊗ Csgn ∼= Csgn ⊕ Csgn−.(d) C[W/〈c〉]⊗ Csgn ∼= C[W/〈c〉].(e) C[W ]⊗ Csgn ∼= C[W ].Proof. For any subgroup H, the character χ of a permutation representation C[W/H]is given byχ(g) = |(W/H)〈g〉|.Ikumitsu NAGASAKI44All isomorphisms are verified by comparing the characters of both sides. By Proposition 2.1, we haveProposition 2.2. Let M be one of W -representations in Proposition 2.1. Then theW -fixed space MW = 0 for cases (II) (a), (b) and (c), and MW = C otherwise.3. Computation by a method of Ariki-Matsuzawa-TeradaLet G = B2, N = NG(T ) and W = N/T as before. We determine dimVN =dim(V T )W for irreducible representation V of G using a method of Ariki-Matsuzawa-Terada [1]. Let V = V (λ) be the irreducible G-representation with the highest weightλ = (λ1, λ2) ∈ Λ+. Let Sk = Sk(C5) be the k-th symmetric power of the natural G-representation on C5. By a result of Weyl [6, p.228, Theorem 7.9A], V (λ) is describedin the representation ring R(G) as follows.Lemma 3.1.V (λ) =∣∣∣∣ Sλ1 − Sλ1−2 Sλ1+1 − Sλ1−3Sλ2−1 − Sλ2−3 Sλ2 − Sλ2−4∣∣∣∣= Sλ1 ⊗ Sλ2 + Sλ1−2 ⊗ Sλ2−4 + Sλ1+1 ⊗ Sλ2−3 + Sλ1−3 ⊗ Sλ2−1− (Sλ1−2 ⊗ Sλ2 + Sλ1 ⊗ Sλ2−4 + Sλ1+1 ⊗ Sλ2−1 + Sλ1−3 ⊗ Sλ2−3) ∈ R(G).In this notation, if k < 0, then Sk is understood to be 0.A basis B of Sμ1 ⊗ Sμ2 is given by the elementsev111 ev122 eα13 eu124 eu115 ⊗ ev211 ev222 eα23 eu224 eu215with vi1+vi2+αi+ui2+ui1 = μi (i = 1, 2). For simplicity, we indicate these elementsby(v1, v2, α, u2, u1) =(v11 v12 α1 u12 u11v21 v22 α2 u22 u21)∈ M2,5(Z≥0).The action of T on Sμ1 ⊗ Sμ2 is given byt · ev111 ev122 eα13 eu124 eu115 ⊗ ev211 ev222 eα23 eu224 eu215= tv11−u11+v21−u211 tv12−u12+v22−u222 ev111 ev122 eα13 eu124 eu115 ⊗ ev211 ev222 eα23 eu224 eu215 .Thus (Sμ1 ⊗ Sμ2)T has a basis BT consisting elements ofev111 ev122 eα13 eu124 eu115 ⊗ ev211 ev222 eα23 eu224 eu215An estimate of the isovariant Borsuk-Ulam constant for a group of type B 2 45satisfying the condition(∗){|vi| := v1i + v2i = |ui| := u1i + u2i (i = 1, 2)vi1 + vi2 + αi + ui2 + ui1 = μi (i = 1, 2).BT is identified with S = S(μ1, μ2) which consists of all elementsv = (v1, v2, α, u2, u1) =(v11 v12 α1 u12 u11v21 v22 α2 u22 u21)∈ M2,5(Z≥0)satisfying the condition (∗).Let a′, b′, c′ be the following elements of the symmetric group SΓ over the setΓ = {±1,±2}:a′ =(1 2 −2 −1−1 2 −2 1), b′ =(1 2 −2 −11 −2 2 −1), c′ =(1 2 −2 −12 1 −1 −2).Let W ′ = 〈a′, b′, c′〉, which is isomorphic to W . The W ′-action on S is given by:a′ · (v1, v2, α, u2, u1) = (u1, v2, α, u2, v1),b′ · (v1, v2, α, u2, u1) = (v1, u2, α, v2, u1),c′ · (v1, v2, α, u2, u1) = (v2, v1, α, u1, u2).Note that these actions are identified with the actions of a, b, c on BT respectively.Hence we obtain a complete system S = S(μ1, μ2) of representatives of S/W′:S = {v = ((v1, v2, α, u2, u1) ∈ S | v1 ≥ u1, v2 ≥ u2, (v1, u1) ≥ (v2, u2)},where(v1, u1) ≥ (v2, u2) ⇐⇒ v1 > v2, or v1 = v2 and u1 ≥ u2.Also set S′ = S′(μ1, μ2) = {v ∈ S |W ′v = 1, 〈c′〉, 〈a′b′c′〉} and S′= S′(μ1, μ2) = S′/W ′.By an easy argument, we haveLemma 3.2. The isotropy subgroup W ′v at v ∈ S is one of following:W ′, 〈a′〉, 〈b′〉, 〈a′, b′〉, 〈c′〉, 〈a′b′c′〉, 1.The result of [1, Theorem B.2] shows that(Sμ1 ⊗ Sμ2)T ∼={⊕v∈SC[W ′/W ′v] (μ1 + μ2 ≡ 0 (2))⊕v∈SC[W ′/W ′v]⊗ Csgn (μ1 + μ2 ≡ 1 (2))Thus, combining this and Propositions 2.1 and 2.2, we haveIkumitsu NAGASAKI46Proposition 3.3.(Sμ1 ⊗ Sμ2)N = ((Sμ1 ⊗ Sμ2)T )W ∼={⊕v∈SC (μ1 + μ2 ≡ 0 (2))⊕v∈S′C (μ1 + μ2 ≡ 1 (2)),and hencedim(Sμ1 ⊗ Sμ2)N ={|S| (μ1 + μ2 ≡ 0 (2))|S′| (μ1 + μ2 ≡ 1 (2)).We give a few examples.Example 3.4. If λ = (1, 0), then V (λ) = S1 ⊗ S0. In this case, S′ = ∅; hencedimV (λ)N = 0.Example 3.5. If λ = (2, 0), then V = V (λ) = S2 ⊗S0 −S0 ⊗S0. In this case, S(2, 0)consists of elements(0 0 2 0 00 0 0 0 0),(1 0 0 0 10 0 0 0 0), and S(0, 0) consists of anelement(0 0 0 0 00 0 0 0 0). Hence we have dimV N = 2− 1 = 1.Example 3.6. If λ = (1, 1), then V = V (λ) = S1 ⊗S1 −S2 ⊗S0. In this case, S(1, 1)consists of elements(0 0 1 0 00 0 1 0 0),(1 0 0 0 00 0 0 0 1), and S(2, 0) consists of twoelements as in the previous example. Hence dimV N = 2− 2 = 0.Let S0(μ1, μ2) for μ1 ≥ 0 and μ2 ≥ 0 be the set of elements of (v1, v2, u2, u1) ∈M2,4(Z≥0) satisfying the condition (∗) with α = 0. For 0 ≤ di ≤ μi, there is aninjection S0(d1, d2) → S(μ1, μ2) defined by(v11 v12 u12 u11v21 v22 u22 u21)→(v11 v12 μ1 − d1 u12 u11v21 v22 μ2 − d2 u22 u21).Then it is easily seen thatLemma 3.7.S(μ1, μ2) ∼=∐0≤d1≤μ10≤d2≤μ2S0(d1, d2),andS′(μ1, μ2) ∼=∐0≤d1≤μ10≤d2≤μ2S′0(d1, d2)as W ′-sets, where S′0(d1, d2) be the subset of S0(d1, d2) consisting of the elements sat-isfying:An estimate of the isovariant Borsuk-Ulam constant for a group of type B 2 47(1) v1 = u1, v2 = u2, and v1 = v2 or u1 = u2, or(2) v1 = v2, u2 = u1, v1 = u1, or(3) v1 = u2, v2 = u1, v1 = v2, where(v1, v2, u2, u1) ∈ M2,4(Z≥0) satisfying the condition (∗) with α = 0.Definition. We setn0(d1, d2) = |S0(d1, d2)/W ′|n′0(d1, d2) = |S′0(d1, d2)/W ′|From condition (∗) with α = 0, it is easy to verify thatLemma 3.8. v ∈ S0(d1, d2) ⇐⇒⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩v11 = −d2 + u11 + v21 + u22 + 2u21v12 = − 12 (d1 + d2)− u11 − v22 − u21u12 =12 (d1 + d2)− u11 − u22 − u21v21 = d2 − v22 − u22 − u210 ≤ uij ≤ di, 0 ≤ vij ≤ di.In particular, we seeCorollary 3.9. If d1 + d2 is odd, then n0(d1, d2) = 0 and n′0(d1, d2) = 0.As a finial result, we obtain the following formula.Theorem 3.10. With the notation above,(1) If λ1 + λ2 is even, then dimV (λ)N =n0(λ1, λ2) + n0(λ1 − 1, λ2 − 3)− n0(λ1 + 1, λ2 − 1)− n0(λ1 − 2, λ2 − 2).(2) If λ1 + λ2 is odd, then dimV (λ)N =n′0(λ1 − 1, λ2) + n′0(λ1, λ2 − 3)− n′0(λ1 + 1, λ2 − 2)− n′0(λ1 − 2, λ2 − 1).Proof. Suppose that λ1 + λ2 is even. By Proposition 3.3, Lemma 3.7 and Corollary3.9, we havedim(Sμ1 ⊗ Sμ2)N =∑0≤d1≤μ10≤d2≤μ2d1+d2≡0 (2)n0(d1, d2).By Lemma 3.1, we obtain statement (1). Statement (2) is similar. Ikumitsu NAGASAKI484. Proof of Theorem 1.1From representation theory of G = B2, recall the dimension formuladimV (λ) =23(λ1 − λ2 + 1)(λ1 + λ2 + 2)(λ1 + 3/2)(λ2 + 1/2),see for example [2]. Suppose that H is a closed subgroup of G with cH = 1. SetrG(H) := sup{dimV (λ)H/ dimV (λ) |λ ∈ Λ+  {0}}.It follows from a similar argument as in [3, 4] that1− rG(H) ≤ cG ≤ 1.Since N is solvable, it follows from [5] that cN = 1. Therefore, it is suffices to showthe following.Proposition 4.1. rG(N) =114 for N = NG(T ).To show this, we first note the following.Lemma 4.2. If λ1 ≥ 11, then dimV (λ)NdimV (λ) <114 .Proof. By [3, Lemma 5.5], we havedimV (λ)NdimV (λ)≤ dimV (λ)TdimV (λ)≤ λ1 + λ2/2Kλ + λ1 + λ2/2,where Kλ = λ21 + λ22 + 3λ1 + λ2. Then the inequalityλ1 + λ2/2Kλ + λ1 + λ2/2<114is equivalent to(λ1 − 5)2 + (λ2 − 114)2 >52116.It is easy to verify that this inequality holds when λ1 ≥ 11. Proof of Proposition 4.1. By Lemma 4.2, we may assume that 0 ≤ λ2 ≤ λ1 ≤ 10((λ1, λ2) = (0, 0)). We compute dimV (λ)N/ dimV (λ) by Theorem 3.10 and the di-mension formula. By computation by Mathematica, we obtain the values ofdimV (λ)N/ dimV (λ)in Table 2 below, and thus we conclude that rG(N) =114. An estimate of the isovariant Borsuk-Ulam constant for a group of type B 2 49λ10 1 2 3 4 5 6 7 8 9 10λ2 0 − 0 114 0 255 0 170 0 195 0 35061 0 0 0 0 1260 01595 01567 02 135110511101390362529352665191014833 0 0 1455177015951172932401116104 21651429127016631399194516445 0 1715113091104533080343016 245521105149433094214957 0 1161522835343708 13232226117829 0 2305910 31771Table 2References[1] A. Ariki, J. Matsuzawa and I. Terada, Representations of Weyl groups on zeroweight spaces of g-modules, Algebraic and Topological Theories (1985) 546–568.[2] T. Bro¨cker and T. tom Dieck, Representations of compact Lie groups, GraduateTexts in Math. 98, Springer 1985.[3] I. Nagasaki, Estimates of the isovariant Borsuk-Ulam constants of compact Liegroups, to appear.[4] I. Nagasaki, Zero weight spaces of irreducible representations and a new estimateof the isovariant Borsuk-Ulam constant of a compact Lie group for SU(3), StudiaHumana et Naturalia 50 (2017), 29–40.[5] A. G. Wasserman, Isovariant maps and the Borsuk-Ulam theorem, Topology Appli.38 (1991) 155–161.[6] H. Weyl, The classical groups, Princeton Univ. Press, 1953.Ikumitsu NAGASAKI50

An estimate of the isovariant Borsuk-Ulam constant for a group of type B2

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An estimate of the isovariant Borsuk-Ulam constant for a group of type B2

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